| L(s) = 1 | − 9·4-s − 9·9-s + 24·11-s + 17·16-s + 184·19-s + 116·29-s − 448·31-s + 81·36-s + 36·41-s − 216·44-s − 49·49-s − 760·59-s + 1.43e3·61-s + 423·64-s − 1.92e3·71-s − 1.65e3·76-s − 1.79e3·79-s + 81·81-s + 2.07e3·89-s − 216·99-s + 92·101-s + 756·109-s − 1.04e3·116-s − 2.23e3·121-s + 4.03e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 1/3·9-s + 0.657·11-s + 0.265·16-s + 2.22·19-s + 0.742·29-s − 2.59·31-s + 3/8·36-s + 0.137·41-s − 0.740·44-s − 1/7·49-s − 1.67·59-s + 3.01·61-s + 0.826·64-s − 3.20·71-s − 2.49·76-s − 2.55·79-s + 1/9·81-s + 2.47·89-s − 0.219·99-s + 0.0906·101-s + 0.664·109-s − 0.835·116-s − 1.67·121-s + 2.92·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.361255116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361255116\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3494 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8130 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11790 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 43414 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 164382 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 270762 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 380 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 718 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 431782 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 960 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 358322 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 896 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 953478 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1038 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1332542 T^{2} + p^{6} T^{4} \) |
| 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
−10.42595755866739323090356494508, −10.25933895455305968310546091819, −9.674178732454685000049996961408, −9.227145245004230714359924725357, −8.887350438553412334698872153535, −8.809394589780622704061818389405, −7.85908873739547548724140952246, −7.66615696543583308656028630782, −7.08702854628003906995801771606, −6.63258564425627448251898906408, −5.88921960399178111643206010912, −5.34578897225955341319423305355, −5.24910831615715771093284791001, −4.38375148537717703945141083609, −4.04907078036679444790608634026, −3.32461736289031074119610053593, −2.97792479975793590736842334955, −1.89776166478418392284370755303, −1.18480633670555135579211835896, −0.40926267412199916161492296753,
0.40926267412199916161492296753, 1.18480633670555135579211835896, 1.89776166478418392284370755303, 2.97792479975793590736842334955, 3.32461736289031074119610053593, 4.04907078036679444790608634026, 4.38375148537717703945141083609, 5.24910831615715771093284791001, 5.34578897225955341319423305355, 5.88921960399178111643206010912, 6.63258564425627448251898906408, 7.08702854628003906995801771606, 7.66615696543583308656028630782, 7.85908873739547548724140952246, 8.809394589780622704061818389405, 8.887350438553412334698872153535, 9.227145245004230714359924725357, 9.674178732454685000049996961408, 10.25933895455305968310546091819, 10.42595755866739323090356494508
