| L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s − 3·11-s − 14-s + 16-s − 2·19-s + 3·20-s + 3·22-s + 6·23-s + 4·25-s + 28-s − 6·29-s − 5·31-s − 32-s + 3·35-s − 2·37-s + 2·38-s − 3·40-s − 6·41-s − 10·43-s − 3·44-s − 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.188·28-s − 1.11·29-s − 0.898·31-s − 0.176·32-s + 0.507·35-s − 0.328·37-s + 0.324·38-s − 0.474·40-s − 0.937·41-s − 1.52·43-s − 0.452·44-s − 0.884·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34473564569659605983322776053, −6.82169296547700737239650468990, −6.04361909054201474789171278264, −5.32602857894135700793174898000, −4.96309314760733004587208328419, −3.65761373500146428259838632217, −2.75787715793218714647486257909, −2.00894247406913823060920569180, −1.40317373749104312020413431195, 0,
1.40317373749104312020413431195, 2.00894247406913823060920569180, 2.75787715793218714647486257909, 3.65761373500146428259838632217, 4.96309314760733004587208328419, 5.32602857894135700793174898000, 6.04361909054201474789171278264, 6.82169296547700737239650468990, 7.34473564569659605983322776053
