| L(s) = 1 | − 2-s + 6·3-s + 8·4-s − 6·6-s + 9·7-s − 7·8-s − 27·9-s + 37·11-s + 48·12-s − 112·13-s − 9·14-s + 63·16-s − 154·17-s + 27·18-s + 70·19-s + 54·21-s − 37·22-s − 267·23-s − 42·24-s + 112·26-s − 378·27-s + 72·28-s − 325·29-s + 12·31-s + 8·32-s + 222·33-s + 154·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s + 4-s − 0.408·6-s + 0.485·7-s − 0.309·8-s − 9-s + 1.01·11-s + 1.15·12-s − 2.38·13-s − 0.171·14-s + 0.984·16-s − 2.19·17-s + 0.353·18-s + 0.845·19-s + 0.561·21-s − 0.358·22-s − 2.42·23-s − 0.357·24-s + 0.844·26-s − 2.69·27-s + 0.485·28-s − 2.08·29-s + 0.0695·31-s + 0.0441·32-s + 1.17·33-s + 0.776·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.07696344826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07696344826\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T + p^{3} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - 7 T^{2} - p^{3} T^{3} - p^{3} T^{4} - p^{6} T^{5} - 7 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 9 T - 617 T^{2} - 108 T^{3} + 341772 T^{4} - 108 p^{3} T^{5} - 617 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T - 1429 T^{2} - 5032 T^{3} + 4235104 T^{4} - 5032 p^{3} T^{5} - 1429 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 112 T + 5542 T^{2} + 292096 T^{3} + 16642027 T^{4} + 292096 p^{3} T^{5} + 5542 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 77 T + 6152 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 35 T + 8868 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 267 T + 33497 T^{2} + 3593286 T^{3} + 412826112 T^{4} + 3593286 p^{3} T^{5} + 33497 p^{6} T^{6} + 267 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 325 T + 42989 T^{2} + 4503850 T^{3} + 752357050 T^{4} + 4503850 p^{3} T^{5} + 42989 p^{6} T^{6} + 325 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T - 29755 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 638 T + 5466 p T^{2} - 638 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 238 T - 65659 T^{2} - 90202 p T^{3} + 3750124 p^{2} T^{4} - 90202 p^{4} T^{5} - 65659 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 97 T - 89513 T^{2} - 5828924 T^{3} + 2716117672 T^{4} - 5828924 p^{3} T^{5} - 89513 p^{6} T^{6} + 97 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 901 T + 402203 T^{2} + 181958752 T^{3} + 72707691052 T^{4} + 181958752 p^{3} T^{5} + 402203 p^{6} T^{6} + 901 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 224 T + 227798 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 85 T - 385531 T^{2} + 1530170 T^{3} + 110592878620 T^{4} + 1530170 p^{3} T^{5} - 385531 p^{6} T^{6} - 85 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 247 T - 94499 T^{2} + 73718138 T^{3} - 41185520126 T^{4} + 73718138 p^{3} T^{5} - 94499 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 606 T - 46787 T^{2} - 113626818 T^{3} - 29494338708 T^{4} - 113626818 p^{3} T^{5} - 46787 p^{6} T^{6} + 606 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 394 T + 654806 T^{2} - 394 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 811 T + 595758 T^{2} - 811 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 840 T - 409226 T^{2} - 108148320 T^{3} + 651861236355 T^{4} - 108148320 p^{3} T^{5} - 409226 p^{6} T^{6} - 840 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 387 T + 661727 T^{2} - 640690884 T^{3} - 150475346808 T^{4} - 640690884 p^{3} T^{5} + 661727 p^{6} T^{6} + 387 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1065 T + 874888 T^{2} + 1065 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1031 T - 886547 T^{2} + 128011022 T^{3} + 1997473632382 T^{4} + 128011022 p^{3} T^{5} - 886547 p^{6} T^{6} + 1031 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.262761809034376118415677436515, −8.050573286754560772490061473300, −7.910755472489568042014558620113, −7.79793623996487434559040462849, −7.78914781165290771645070955015, −7.07009410369795675174641695155, −6.98490219050015745506896519217, −6.56694936494609542919417219350, −6.38438234768128452042363933867, −6.08905854487404015798892050070, −5.72163526004252281067611996584, −5.55578797917333274848042374668, −4.96254047706967184036661826350, −4.93552538492197080143285347099, −4.18785495279735504264892554070, −4.16338526774365846917216523538, −3.83055255469441570794493907080, −3.37032961288782841098356554640, −2.79330741214283807549331319971, −2.47372598937004086693397737244, −2.45932679546790290488336016700, −2.03102613164662391466985678351, −1.72468330353516380338735233105, −0.917709223741442636192554979380, −0.04909348965796730914412840664,
0.04909348965796730914412840664, 0.917709223741442636192554979380, 1.72468330353516380338735233105, 2.03102613164662391466985678351, 2.45932679546790290488336016700, 2.47372598937004086693397737244, 2.79330741214283807549331319971, 3.37032961288782841098356554640, 3.83055255469441570794493907080, 4.16338526774365846917216523538, 4.18785495279735504264892554070, 4.93552538492197080143285347099, 4.96254047706967184036661826350, 5.55578797917333274848042374668, 5.72163526004252281067611996584, 6.08905854487404015798892050070, 6.38438234768128452042363933867, 6.56694936494609542919417219350, 6.98490219050015745506896519217, 7.07009410369795675174641695155, 7.78914781165290771645070955015, 7.79793623996487434559040462849, 7.910755472489568042014558620113, 8.050573286754560772490061473300, 8.262761809034376118415677436515
