L(s) = 1 | − 2-s − 3·3-s + 4·4-s + 8·5-s + 3·6-s − 10·7-s − 5·8-s + 3·9-s − 8·10-s + 11-s − 12·12-s + 10·14-s − 24·15-s + 10·16-s + 7·17-s − 3·18-s − 11·19-s + 32·20-s + 30·21-s − 22-s − 2·23-s + 15·24-s + 12·25-s + 2·27-s − 40·28-s + 8·29-s + 24·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 2·4-s + 3.57·5-s + 1.22·6-s − 3.77·7-s − 1.76·8-s + 9-s − 2.52·10-s + 0.301·11-s − 3.46·12-s + 2.67·14-s − 6.19·15-s + 5/2·16-s + 1.69·17-s − 0.707·18-s − 2.52·19-s + 7.15·20-s + 6.54·21-s − 0.213·22-s − 0.417·23-s + 3.06·24-s + 12/5·25-s + 0.384·27-s − 7.55·28-s + 1.48·29-s + 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.289720390\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289720390\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T - 3 T^{2} - p T^{3} + 5 T^{4} - 11 T^{6} + 5 p^{2} T^{8} - p^{4} T^{9} - 3 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( ( 1 - 4 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 10 T + 48 T^{2} + 26 p T^{3} + 650 T^{4} + 1998 T^{5} + 5419 T^{6} + 1998 p T^{7} + 650 p^{2} T^{8} + 26 p^{4} T^{9} + 48 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - T - 2 T^{2} - 45 T^{3} - 3 T^{4} + 8 T^{5} + 2883 T^{6} + 8 p T^{7} - 3 p^{2} T^{8} - 45 p^{3} T^{9} - 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 7 T - 16 T^{2} + 35 T^{3} + 1405 T^{4} - 2478 T^{5} - 16135 T^{6} - 2478 p T^{7} + 1405 p^{2} T^{8} + 35 p^{3} T^{9} - 16 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 11 T + 54 T^{2} + 127 T^{3} - 139 T^{4} - 3600 T^{5} - 22229 T^{6} - 3600 p T^{7} - 139 p^{2} T^{8} + 127 p^{3} T^{9} + 54 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 2 T - 22 T^{2} - 298 T^{3} - 272 T^{4} + 3216 T^{5} + 37295 T^{6} + 3216 p T^{7} - 272 p^{2} T^{8} - 298 p^{3} T^{9} - 22 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 8 T - 28 T^{2} + 106 T^{3} + 2428 T^{4} - 2772 T^{5} - 73261 T^{6} - 2772 p T^{7} + 2428 p^{2} T^{8} + 106 p^{3} T^{9} - 28 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 8 T + 70 T^{2} - 299 T^{3} + 70 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 14 T + 22 T^{2} + 182 T^{3} + 8060 T^{4} + 35000 T^{5} - 17101 T^{6} + 35000 p T^{7} + 8060 p^{2} T^{8} + 182 p^{3} T^{9} + 22 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + T - 120 T^{2} - p T^{3} + 9599 T^{4} + 1560 T^{5} - 457559 T^{6} + 1560 p T^{7} + 9599 p^{2} T^{8} - p^{4} T^{9} - 120 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 95 T^{2} + 262 T^{3} + 5483 T^{4} - 8563 T^{5} - 240346 T^{6} - 8563 p T^{7} + 5483 p^{2} T^{8} + 262 p^{3} T^{9} - 95 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 9 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 14 T + 19 T^{2} + 714 T^{3} - 1458 T^{4} - 658 p T^{5} + 427287 T^{6} - 658 p^{2} T^{7} - 1458 p^{2} T^{8} + 714 p^{3} T^{9} + 19 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 13 T - 26 T^{2} + 191 T^{3} + 8411 T^{4} - 4888 T^{5} - 636643 T^{6} - 4888 p T^{7} + 8411 p^{2} T^{8} + 191 p^{3} T^{9} - 26 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 5 T - 154 T^{2} - 251 T^{3} + 17183 T^{4} + 5082 T^{5} - 1329117 T^{6} + 5082 p T^{7} + 17183 p^{2} T^{8} - 251 p^{3} T^{9} - 154 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 6 T - 98 T^{2} - 22 T^{3} + 6096 T^{4} + 31528 T^{5} - 587649 T^{6} + 31528 p T^{7} + 6096 p^{2} T^{8} - 22 p^{3} T^{9} - 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 18 T + 320 T^{2} - 2795 T^{3} + 320 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 16 T + 304 T^{2} + 2613 T^{3} + 304 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 5 T - 234 T^{2} + 487 T^{3} + 39575 T^{4} - 39864 T^{5} - 3964415 T^{6} - 39864 p T^{7} + 39575 p^{2} T^{8} + 487 p^{3} T^{9} - 234 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 5 T - p T^{2} + 1048 T^{3} + 4145 T^{4} - 98013 T^{5} + 194334 T^{6} - 98013 p T^{7} + 4145 p^{2} T^{8} + 1048 p^{3} T^{9} - p^{5} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.99439522831110963680160335571, −5.88736993533440203811861275889, −5.71588136804552761438193331129, −5.61839355721030471794321784695, −5.56228729302144049580540683343, −5.08181199096288147535901230731, −4.99246656155158216296576287685, −4.84482290737038705335119133066, −4.54938569204671455149643794183, −4.28373939076048124009946217453, −3.81405578948723405946821624669, −3.77419269358230246765221701420, −3.67913489503066777145453480677, −3.21219233542648758830026458624, −3.08130724156976215944612025761, −2.99068763832104572656591770281, −2.83247044062705538197664831192, −2.31395479637444609579567994587, −2.30910479003689922645922518991, −2.14359875952005769838066149262, −1.74987795426988844687777092534, −1.68233379865755431224007563826, −1.33715567769324276998204945654, −0.54868188871355063865016741664, −0.44972993071087447733884193620,
0.44972993071087447733884193620, 0.54868188871355063865016741664, 1.33715567769324276998204945654, 1.68233379865755431224007563826, 1.74987795426988844687777092534, 2.14359875952005769838066149262, 2.30910479003689922645922518991, 2.31395479637444609579567994587, 2.83247044062705538197664831192, 2.99068763832104572656591770281, 3.08130724156976215944612025761, 3.21219233542648758830026458624, 3.67913489503066777145453480677, 3.77419269358230246765221701420, 3.81405578948723405946821624669, 4.28373939076048124009946217453, 4.54938569204671455149643794183, 4.84482290737038705335119133066, 4.99246656155158216296576287685, 5.08181199096288147535901230731, 5.56228729302144049580540683343, 5.61839355721030471794321784695, 5.71588136804552761438193331129, 5.88736993533440203811861275889, 5.99439522831110963680160335571
Plot not available for L-functions of degree greater than 10.