| L(s) = 1 | + 14·7-s + 114·9-s − 36·11-s − 1.47e3·23-s + 818·25-s − 1.69e3·29-s + 4.77e3·37-s + 5.02e3·43-s − 2.20e3·49-s − 540·53-s + 1.59e3·63-s − 4.90e3·67-s + 6.30e3·71-s − 504·77-s + 7.96e3·79-s + 6.43e3·81-s − 4.10e3·99-s + 2.58e4·107-s − 1.40e4·109-s + 3.74e4·113-s − 2.83e4·121-s + ⋯ |
| L(s) = 1 | + 2/7·7-s + 1.40·9-s − 0.297·11-s − 2.79·23-s + 1.30·25-s − 2.01·29-s + 3.48·37-s + 2.71·43-s − 0.918·49-s − 0.192·53-s + 0.402·63-s − 1.09·67-s + 1.24·71-s − 0.0850·77-s + 1.27·79-s + 0.980·81-s − 0.418·99-s + 2.26·107-s − 1.18·109-s + 2.93·113-s − 1.93·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.489284462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489284462\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 p T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 818 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 18 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 39794 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5758 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 252530 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 738 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 846 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 492290 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2386 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1634 p^{2} T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2510 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1859710 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 270 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14384690 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14450830 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2450 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3150 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56740994 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3982 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 69825650 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 67615490 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18761474 T^{2} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.93997637496691517006044544293, −12.80542608475762512480276720991, −12.28505991177933616882799617872, −11.53673190360222920459203613827, −11.06236510817886006001609349216, −10.58574090065378163520661760306, −9.857087987980905322922197116658, −9.598859300611313376729009545953, −9.009027270915914457101486548318, −8.032488596217999650160949835652, −7.67209360208109298291859642127, −7.33741622320023770826533690824, −6.20193349317751239318678132210, −6.03067691849715852510849060060, −4.98442354356402267929762590112, −4.22011040149898411990694176674, −3.90129006039695072896676767604, −2.56239798627585230310104891607, −1.79377021411836531981096570147, −0.71911200776598591929907872419,
0.71911200776598591929907872419, 1.79377021411836531981096570147, 2.56239798627585230310104891607, 3.90129006039695072896676767604, 4.22011040149898411990694176674, 4.98442354356402267929762590112, 6.03067691849715852510849060060, 6.20193349317751239318678132210, 7.33741622320023770826533690824, 7.67209360208109298291859642127, 8.032488596217999650160949835652, 9.009027270915914457101486548318, 9.598859300611313376729009545953, 9.857087987980905322922197116658, 10.58574090065378163520661760306, 11.06236510817886006001609349216, 11.53673190360222920459203613827, 12.28505991177933616882799617872, 12.80542608475762512480276720991, 12.93997637496691517006044544293
