L(s) = 1 | − 4·3-s + 8·9-s − 96·13-s − 16·19-s + 12·27-s − 72·31-s + 112·37-s + 384·39-s + 240·43-s + 360·49-s + 64·57-s + 208·61-s + 232·67-s + 136·79-s − 34·81-s + 288·93-s − 480·97-s + 64·109-s − 448·111-s − 768·117-s + 127-s − 960·129-s + 131-s + 137-s + 139-s − 1.44e3·147-s + 149-s + ⋯ |
L(s) = 1 | − 4/3·3-s + 8/9·9-s − 7.38·13-s − 0.842·19-s + 4/9·27-s − 2.32·31-s + 3.02·37-s + 9.84·39-s + 5.58·43-s + 7.34·49-s + 1.12·57-s + 3.40·61-s + 3.46·67-s + 1.72·79-s − 0.419·81-s + 3.09·93-s − 4.94·97-s + 0.587·109-s − 4.03·111-s − 6.56·117-s + 0.00787·127-s − 7.44·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 9.79·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.625241531\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.625241531\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T + 8 T^{2} - 4 p T^{3} - 14 p^{2} T^{4} - 4 p^{3} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
good | 5 | \( 1 + 372 T^{4} + 464614 T^{8} + 372 p^{8} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 180 T^{2} + 12874 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 37380 T^{4} + 692870854 T^{8} + 37380 p^{8} T^{12} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 48 T + 1152 T^{2} + 21264 T^{3} + 317422 T^{4} + 21264 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 236 T^{2} + 73334 T^{4} - 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 8 T + 32 T^{2} + 1160 T^{3} - 4606 T^{4} + 1160 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 996 T^{2} + 719878 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 147060 T^{4} + 623812106854 T^{8} + 147060 p^{8} T^{12} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 18 T + 1996 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 56 T + 1568 T^{2} - 79016 T^{3} + 3980078 T^{4} - 79016 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6060 T^{2} + 14724790 T^{4} + 6060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 120 T + 7200 T^{2} - 411000 T^{3} + 20977474 T^{4} - 411000 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 692 T^{2} + 7491686 T^{4} - 692 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 + 1441524 T^{4} - 91064986479386 T^{8} + 1441524 p^{8} T^{12} + p^{16} T^{16} \) |
| 59 | \( 1 + 15787044 T^{4} + 346992256453126 T^{8} + 15787044 p^{8} T^{12} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 104 T + 5408 T^{2} - 491192 T^{3} + 43609454 T^{4} - 491192 p^{2} T^{5} + 5408 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 116 T + 6728 T^{2} - 350436 T^{3} + 16097858 T^{4} - 350436 p^{2} T^{5} + 6728 p^{4} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 3604 T^{2} + 10180454 T^{4} + 3604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17604 T^{2} + 131615494 T^{4} - 17604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 34 T + 11588 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( 1 - 64624380 T^{4} + 2364945173919814 T^{8} - 64624380 p^{8} T^{12} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 25444 T^{2} + 277784198 T^{4} + 25444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 120 T + 21970 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.40473440215441547316529739911, −5.39514927328910254632705294268, −5.21362026340395800426310175921, −5.16157315134090203603686326730, −4.68861510215831280088986561437, −4.68777704979338337160627900663, −4.67586827216423457361549689884, −4.19622509605699924584103702617, −4.13952042736782982274595913291, −4.11942126608129602738527126101, −4.06399266266121843652178026243, −3.71255868839908112490763650475, −3.64130813587629485274942611988, −2.88190735855970880694419593674, −2.69291988703037363756764567971, −2.64323368347480831200107615349, −2.55059706610198536308316647099, −2.34632110917814461012773088662, −2.27713030241238918261971089269, −2.26395594157729484041766228028, −1.83654079841516906502466647365, −0.947615677416685273639975781063, −0.76972115015039227886187311764, −0.53285264370671593875849076703, −0.41422145560780913823239684336,
0.41422145560780913823239684336, 0.53285264370671593875849076703, 0.76972115015039227886187311764, 0.947615677416685273639975781063, 1.83654079841516906502466647365, 2.26395594157729484041766228028, 2.27713030241238918261971089269, 2.34632110917814461012773088662, 2.55059706610198536308316647099, 2.64323368347480831200107615349, 2.69291988703037363756764567971, 2.88190735855970880694419593674, 3.64130813587629485274942611988, 3.71255868839908112490763650475, 4.06399266266121843652178026243, 4.11942126608129602738527126101, 4.13952042736782982274595913291, 4.19622509605699924584103702617, 4.67586827216423457361549689884, 4.68777704979338337160627900663, 4.68861510215831280088986561437, 5.16157315134090203603686326730, 5.21362026340395800426310175921, 5.39514927328910254632705294268, 5.40473440215441547316529739911
Plot not available for L-functions of degree greater than 10.