| L(s) = 1 | − 4·4-s + 38·9-s − 88·11-s + 16·16-s − 38·19-s + 324·29-s − 624·31-s − 152·36-s + 68·41-s + 352·44-s + 286·49-s − 728·59-s + 1.03e3·61-s − 64·64-s + 640·71-s + 152·76-s + 2.41e3·79-s + 715·81-s + 2.04e3·89-s − 3.34e3·99-s − 1.79e3·101-s − 2.09e3·109-s − 1.29e3·116-s + 3.14e3·121-s + 2.49e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.40·9-s − 2.41·11-s + 1/4·16-s − 0.458·19-s + 2.07·29-s − 3.61·31-s − 0.703·36-s + 0.259·41-s + 1.20·44-s + 0.833·49-s − 1.60·59-s + 2.17·61-s − 1/8·64-s + 1.06·71-s + 0.229·76-s + 3.44·79-s + 0.980·81-s + 2.43·89-s − 3.39·99-s − 1.76·101-s − 1.83·109-s − 1.03·116-s + 2.36·121-s + 1.80·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.312217700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312217700\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2430 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2562 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 27610 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 128754 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 41718 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 364 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 252250 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 484270 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1208 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 110826 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1022 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 465790 T^{2} + p^{6} T^{4} \) |
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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
−9.882721140005250697826044375749, −9.401091254797735589634204044887, −9.183769785699725115909318318797, −8.581470023089136208719360511137, −8.062665340537514203129447348305, −7.75009936945931162017649807694, −7.55368864538176865495321368123, −6.78130466521080935894303728682, −6.72177131456271208666227611312, −5.82886638171804293756282590521, −5.42394012778622198073007603905, −4.91598918117742613558666823765, −4.84685854497597558147563005285, −3.92651949796314032549333748919, −3.75363955329574058965639566199, −2.96897344424761143363456060342, −2.29956341570017540144513040906, −1.95274250460059837051992414129, −1.01310994925174928615743872480, −0.33415302523609919438880253279,
0.33415302523609919438880253279, 1.01310994925174928615743872480, 1.95274250460059837051992414129, 2.29956341570017540144513040906, 2.96897344424761143363456060342, 3.75363955329574058965639566199, 3.92651949796314032549333748919, 4.84685854497597558147563005285, 4.91598918117742613558666823765, 5.42394012778622198073007603905, 5.82886638171804293756282590521, 6.72177131456271208666227611312, 6.78130466521080935894303728682, 7.55368864538176865495321368123, 7.75009936945931162017649807694, 8.062665340537514203129447348305, 8.581470023089136208719360511137, 9.183769785699725115909318318797, 9.401091254797735589634204044887, 9.882721140005250697826044375749
