| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 8-s + 9-s + 2·10-s − 4·11-s − 2·12-s − 2·13-s − 4·15-s + 16-s + 2·17-s + 18-s + 2·20-s − 4·22-s − 2·24-s − 25-s − 2·26-s + 4·27-s + 8·29-s − 4·30-s − 8·31-s + 32-s + 8·33-s + 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.852·22-s − 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s + 1.48·29-s − 0.730·30-s − 1.43·31-s + 0.176·32-s + 1.39·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.41247665800079017445564780764, −6.41621134075799175382990858972, −6.06054418737238962273469870406, −5.30160251066217533978538036837, −5.06291605340298407804852356513, −4.18256462257521244802954644657, −2.97350971705358375200834260484, −2.40700628095474540106882097356, −1.29716680922222776935948076744, 0,
1.29716680922222776935948076744, 2.40700628095474540106882097356, 2.97350971705358375200834260484, 4.18256462257521244802954644657, 5.06291605340298407804852356513, 5.30160251066217533978538036837, 6.06054418737238962273469870406, 6.41621134075799175382990858972, 7.41247665800079017445564780764
