| L(s) = 1 | − 32·2-s − 114·3-s + 512·4-s + 3.64e3·6-s − 1.39e4·7-s + 6.49e3·9-s + 1.50e5·11-s − 5.83e4·12-s − 2.19e5·13-s + 4.44e5·14-s − 2.62e5·16-s + 3.05e6·17-s − 2.07e5·18-s + 1.58e6·21-s − 4.81e6·22-s + 1.42e6·23-s + 7.03e6·26-s − 6.73e6·27-s − 7.11e6·28-s − 5.81e7·31-s + 8.38e6·32-s − 1.71e7·33-s − 9.78e7·34-s + 3.32e6·36-s + 1.82e6·37-s + 2.50e7·39-s − 3.27e8·41-s + ⋯ |
| L(s) = 1 | − 2-s − 0.469·3-s + 1/2·4-s + 0.469·6-s − 0.827·7-s + 0.110·9-s + 0.934·11-s − 0.234·12-s − 0.591·13-s + 0.827·14-s − 1/4·16-s + 2.15·17-s − 0.110·18-s + 0.388·21-s − 0.934·22-s + 0.221·23-s + 0.591·26-s − 0.469·27-s − 0.413·28-s − 2.03·31-s + 1/4·32-s − 0.438·33-s − 2.15·34-s + 0.0550·36-s + 0.0262·37-s + 0.277·39-s − 2.83·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.05408632950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05408632950\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 + p^{5} T + p^{9} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 38 p T + 722 p^{2} T^{2} + 38 p^{11} T^{3} + p^{20} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 13906 T + 96688418 T^{2} + 13906 p^{10} T^{3} + p^{20} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 75242 T + p^{10} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 219714 T + 24137120898 T^{2} + 219714 p^{10} T^{3} + p^{20} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3057854 T + 4675235542658 T^{2} - 3057854 p^{10} T^{3} + p^{20} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4048803626798 T^{2} + p^{20} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1424846 T + 1015093061858 T^{2} - 1424846 p^{10} T^{3} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 841215443546002 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 29080718 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 1823694 T + 1662929902818 T^{2} - 1823694 p^{10} T^{3} + p^{20} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 163945678 T + p^{10} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 236845554 T + 28047908224783458 T^{2} + 236845554 p^{10} T^{3} + p^{20} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 552640626 T + 152705830752835938 T^{2} + 552640626 p^{10} T^{3} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 616920194 T + 190295262882498818 T^{2} + 616920194 p^{10} T^{3} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 136962600617793202 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1353610038 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1707141826 T + 1457166607039307138 T^{2} + 1707141826 p^{10} T^{3} + p^{20} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2827014562 T + p^{10} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5506594366 T + 15161290755831470978 T^{2} - 5506594366 p^{10} T^{3} + p^{20} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7884247659893284802 T^{2} + p^{20} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2692678194 T + 3625257928221550818 T^{2} + 2692678194 p^{10} T^{3} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 55248148650264264802 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1053125694 T + 554536863681490818 T^{2} - 1053125694 p^{10} T^{3} + p^{20} T^{4} \) |
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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
−14.18372494345407568817938000009, −12.73073995519417771434369496251, −12.56326343675109957221452254038, −11.72771793285815771583621478853, −11.43270892380620215613831308383, −10.48240936407406714423069113323, −10.04754014541284730538026647482, −9.461173932436742630545809215987, −9.186562417435248142565010409096, −8.077214290239302451797969462464, −7.74809567368662314376075507510, −6.74174633642814811318976122084, −6.51185684613515801034109924910, −5.43988371322632669965077383488, −4.92269609493707396922001017704, −3.41492483558793365182031605772, −3.38430457620361429325875811244, −1.74019643026218169195539842455, −1.34201803750423587427678624208, −0.091638243301735977552465553434,
0.091638243301735977552465553434, 1.34201803750423587427678624208, 1.74019643026218169195539842455, 3.38430457620361429325875811244, 3.41492483558793365182031605772, 4.92269609493707396922001017704, 5.43988371322632669965077383488, 6.51185684613515801034109924910, 6.74174633642814811318976122084, 7.74809567368662314376075507510, 8.077214290239302451797969462464, 9.186562417435248142565010409096, 9.461173932436742630545809215987, 10.04754014541284730538026647482, 10.48240936407406714423069113323, 11.43270892380620215613831308383, 11.72771793285815771583621478853, 12.56326343675109957221452254038, 12.73073995519417771434369496251, 14.18372494345407568817938000009
