L(s) = 1 | + 3·4-s + 2·5-s + 12·11-s + 5·16-s − 12·19-s + 6·20-s − 25-s − 4·29-s + 20·31-s + 4·41-s + 36·44-s + 24·55-s + 16·59-s + 4·61-s + 3·64-s − 20·71-s − 36·76-s − 8·79-s + 10·80-s − 12·89-s − 24·95-s − 3·100-s − 12·101-s − 4·109-s − 12·116-s + 86·121-s + 60·124-s + ⋯ |
L(s) = 1 | + 3/2·4-s + 0.894·5-s + 3.61·11-s + 5/4·16-s − 2.75·19-s + 1.34·20-s − 1/5·25-s − 0.742·29-s + 3.59·31-s + 0.624·41-s + 5.42·44-s + 3.23·55-s + 2.08·59-s + 0.512·61-s + 3/8·64-s − 2.37·71-s − 4.12·76-s − 0.900·79-s + 1.11·80-s − 1.27·89-s − 2.46·95-s − 0.299·100-s − 1.19·101-s − 0.383·109-s − 1.11·116-s + 7.81·121-s + 5.38·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.420668986\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.420668986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.152015619096937725207148203158, −8.926832819636393698701746971037, −8.412928316515727147529402649335, −8.324086314088260903286211992260, −7.61913039043795394897084926451, −6.94374272381862301057236724196, −6.74289647431043218547640321919, −6.59765281391358485624757472496, −6.27469358068490926207003117890, −5.86039425343849187090355705307, −5.70687740042207680765704750497, −4.49555301422447838119620216232, −4.43550523097462089520048091438, −4.00056205547625356527093407031, −3.55116177053758679806291577369, −2.68955168677024235883511577316, −2.48261377113560904590093539749, −1.75625004967303957429671382069, −1.54357849409318562023686122849, −0.910232915939716783487750563057,
0.910232915939716783487750563057, 1.54357849409318562023686122849, 1.75625004967303957429671382069, 2.48261377113560904590093539749, 2.68955168677024235883511577316, 3.55116177053758679806291577369, 4.00056205547625356527093407031, 4.43550523097462089520048091438, 4.49555301422447838119620216232, 5.70687740042207680765704750497, 5.86039425343849187090355705307, 6.27469358068490926207003117890, 6.59765281391358485624757472496, 6.74289647431043218547640321919, 6.94374272381862301057236724196, 7.61913039043795394897084926451, 8.324086314088260903286211992260, 8.412928316515727147529402649335, 8.926832819636393698701746971037, 9.152015619096937725207148203158