| L(s) = 1 | − 3-s + 4-s − 8·5-s + 3·7-s − 2·9-s − 12-s + 8·15-s + 16-s + 6·17-s − 8·20-s − 3·21-s + 38·25-s + 5·27-s + 3·28-s − 24·35-s − 2·36-s − 4·37-s − 16·41-s + 8·43-s + 16·45-s + 16·47-s − 48-s + 2·49-s − 6·51-s + 30·59-s + 8·60-s − 6·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 3.57·5-s + 1.13·7-s − 2/3·9-s − 0.288·12-s + 2.06·15-s + 1/4·16-s + 1.45·17-s − 1.78·20-s − 0.654·21-s + 38/5·25-s + 0.962·27-s + 0.566·28-s − 4.05·35-s − 1/3·36-s − 0.657·37-s − 2.49·41-s + 1.21·43-s + 2.38·45-s + 2.33·47-s − 0.144·48-s + 2/7·49-s − 0.840·51-s + 3.90·59-s + 1.03·60-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7322993137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7322993137\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.407330577817794343806946011608, −8.023773930245372820786799880627, −7.26870555945267237162800864197, −7.25038064772684706053554512263, −7.01458996663323080185675062091, −6.09321920626294409315899202682, −5.28069074518443602737935578596, −5.26436680711797090374689452576, −4.54613755561444218090148036459, −3.86057015395193111701613139713, −3.85664265110658888269329549605, −3.14464911774177170032818923318, −2.57437731667109469936792684003, −1.21442896549369149694295509296, −0.51687700273278494466792347987,
0.51687700273278494466792347987, 1.21442896549369149694295509296, 2.57437731667109469936792684003, 3.14464911774177170032818923318, 3.85664265110658888269329549605, 3.86057015395193111701613139713, 4.54613755561444218090148036459, 5.26436680711797090374689452576, 5.28069074518443602737935578596, 6.09321920626294409315899202682, 7.01458996663323080185675062091, 7.25038064772684706053554512263, 7.26870555945267237162800864197, 8.023773930245372820786799880627, 8.407330577817794343806946011608
