| L(s) = 1 | + 2·3-s + 4-s − 6·7-s + 9-s + 6·11-s + 2·12-s − 4·13-s − 6·19-s − 12·21-s − 6·23-s + 14·25-s − 2·27-s − 6·28-s + 6·29-s + 12·33-s + 36-s − 8·39-s + 12·41-s + 2·43-s + 6·44-s + 16·49-s − 4·52-s + 12·53-s − 12·57-s − 48·59-s − 20·61-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 2.26·7-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 1.10·13-s − 1.37·19-s − 2.61·21-s − 1.25·23-s + 14/5·25-s − 0.384·27-s − 1.13·28-s + 1.11·29-s + 2.08·33-s + 1/6·36-s − 1.28·39-s + 1.87·41-s + 0.304·43-s + 0.904·44-s + 16/7·49-s − 0.554·52-s + 1.64·53-s − 1.58·57-s − 6.24·59-s − 2.56·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9854868284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9854868284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 20 T^{2} + 48 T^{3} + 99 T^{4} + 48 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 133 T^{2} - 1020 T^{3} + 7512 T^{4} - 1020 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 52 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 24 T + 251 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 68 T^{2} + 336 T^{3} - 549 T^{4} + 336 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−10.85202199386372656451985597941, −10.33620147586997298665821387731, −10.17494998809962064716079754732, −9.664680484433959991954181692700, −9.552376313823090477480147145279, −9.184375554998852900409044871602, −8.950812568288712034051387321784, −8.917102257824684048205035529725, −8.489836018754490918413313919452, −7.904887659821208145754925150705, −7.73448658827707220274435635388, −7.17877352072917162392944041289, −7.10699435434673673449135277964, −6.56786227303523526596374036781, −6.26484740831525931551073563134, −6.15874339313983288500300342785, −6.03580985518465871879523139507, −4.98937760909974793030923450180, −4.67328986939593676657511962972, −4.05794074943596429287036818958, −4.01113955825400049886541738745, −3.10639155598610243462545224339, −2.83643051103933429250281585704, −2.78365226814312808697877704458, −1.70157047061208495340155103517,
1.70157047061208495340155103517, 2.78365226814312808697877704458, 2.83643051103933429250281585704, 3.10639155598610243462545224339, 4.01113955825400049886541738745, 4.05794074943596429287036818958, 4.67328986939593676657511962972, 4.98937760909974793030923450180, 6.03580985518465871879523139507, 6.15874339313983288500300342785, 6.26484740831525931551073563134, 6.56786227303523526596374036781, 7.10699435434673673449135277964, 7.17877352072917162392944041289, 7.73448658827707220274435635388, 7.904887659821208145754925150705, 8.489836018754490918413313919452, 8.917102257824684048205035529725, 8.950812568288712034051387321784, 9.184375554998852900409044871602, 9.552376313823090477480147145279, 9.664680484433959991954181692700, 10.17494998809962064716079754732, 10.33620147586997298665821387731, 10.85202199386372656451985597941
