| L(s) = 1 | − 2·2-s − 4-s + 8·8-s + 9-s − 4·13-s − 7·16-s + 2·17-s − 2·18-s + 8·19-s + 25-s + 8·26-s − 14·32-s − 4·34-s − 36-s − 16·38-s + 8·43-s + 16·47-s − 14·49-s − 2·50-s + 4·52-s − 20·53-s − 8·59-s + 35·64-s + 24·67-s − 2·68-s + 8·72-s − 8·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1/3·9-s − 1.10·13-s − 7/4·16-s + 0.485·17-s − 0.471·18-s + 1.83·19-s + 1/5·25-s + 1.56·26-s − 2.47·32-s − 0.685·34-s − 1/6·36-s − 2.59·38-s + 1.21·43-s + 2.33·47-s − 2·49-s − 0.282·50-s + 0.554·52-s − 2.74·53-s − 1.04·59-s + 35/8·64-s + 2.93·67-s − 0.242·68-s + 0.942·72-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4757794391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4757794391\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.523451675812284265792380668213, −9.433270536163240511450818045276, −9.275278471815784671366808718367, −8.324027666892170998148797227210, −7.88272580417467619012367098000, −7.66488013441745380243230842523, −7.16750910474070634391114159922, −6.44628802162953221998351414777, −5.26301391890678078722159871955, −5.23920392624592057055772361749, −4.47629935336188211298931611555, −3.84194557004788343759428710030, −2.95453421762719696432427292949, −1.69452749631301531601828409816, −0.75602502315476554931106499220,
0.75602502315476554931106499220, 1.69452749631301531601828409816, 2.95453421762719696432427292949, 3.84194557004788343759428710030, 4.47629935336188211298931611555, 5.23920392624592057055772361749, 5.26301391890678078722159871955, 6.44628802162953221998351414777, 7.16750910474070634391114159922, 7.66488013441745380243230842523, 7.88272580417467619012367098000, 8.324027666892170998148797227210, 9.275278471815784671366808718367, 9.433270536163240511450818045276, 9.523451675812284265792380668213
