| L(s) = 1 | − 2·2-s + 6·3-s + 6·4-s − 12·6-s + 26·7-s − 32·8-s + 27·9-s + 28·11-s + 36·12-s + 18·13-s − 52·14-s + 44·16-s − 68·17-s − 54·18-s + 6·19-s + 156·21-s − 56·22-s + 132·23-s − 192·24-s − 36·26-s + 108·27-s + 156·28-s + 92·29-s + 122·31-s − 120·32-s + 168·33-s + 136·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 3/4·4-s − 0.816·6-s + 1.40·7-s − 1.41·8-s + 9-s + 0.767·11-s + 0.866·12-s + 0.384·13-s − 0.992·14-s + 0.687·16-s − 0.970·17-s − 0.707·18-s + 0.0724·19-s + 1.62·21-s − 0.542·22-s + 1.19·23-s − 1.63·24-s − 0.271·26-s + 0.769·27-s + 1.05·28-s + 0.589·29-s + 0.706·31-s − 0.662·32-s + 0.886·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.493124949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.493124949\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T - p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 26 T + 551 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2554 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 18 T - 389 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 p T + 3382 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 13423 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 132 T + 25954 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 92 T + 35998 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 122 T + 48407 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 284 T + 110526 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 392 T + 124882 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 690 T + 277735 T^{2} - 690 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 620 T + 284290 T^{2} + 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 p T + 462634 T^{2} + 16 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 124 T + 336778 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 750 T + 535003 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 358 T + 443567 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 824 T + 877966 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 108 T + 779734 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 880 T + 693278 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 156 T + 449242 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 864 T + 1202578 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 521 T + p^{3} T^{2} )^{2} \) |
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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
−14.29167349720641138117828163268, −14.24332550083983904882982269161, −13.18550028868367366655362509949, −12.74100189325988967362544988474, −12.01807971394803128186190110926, −11.31282448775903488562453309718, −11.15476230262807161802078519116, −10.42879373237086755428567101908, −9.358852671101762233775730606498, −9.297601093707357729820412345387, −8.509584476741105792359814000800, −8.260820253373143265103536819552, −7.47469207047706828605745659250, −6.77615100985806070963595453584, −6.20796267429547389008604065886, −5.00215006540971788479692198185, −4.20267866471486261462943912769, −3.11584581666069928201061840544, −2.26218906252631476550710811291, −1.23234670314887101005053616116,
1.23234670314887101005053616116, 2.26218906252631476550710811291, 3.11584581666069928201061840544, 4.20267866471486261462943912769, 5.00215006540971788479692198185, 6.20796267429547389008604065886, 6.77615100985806070963595453584, 7.47469207047706828605745659250, 8.260820253373143265103536819552, 8.509584476741105792359814000800, 9.297601093707357729820412345387, 9.358852671101762233775730606498, 10.42879373237086755428567101908, 11.15476230262807161802078519116, 11.31282448775903488562453309718, 12.01807971394803128186190110926, 12.74100189325988967362544988474, 13.18550028868367366655362509949, 14.24332550083983904882982269161, 14.29167349720641138117828163268
