| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 4·11-s − 2·12-s − 2·13-s + 16-s + 2·17-s − 18-s + 4·22-s + 2·24-s + 2·26-s + 4·27-s + 8·29-s + 8·31-s − 32-s + 8·33-s − 2·34-s + 36-s − 4·37-s + 4·39-s − 10·41-s + 2·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.852·22-s + 0.408·24-s + 0.392·26-s + 0.769·27-s + 1.48·29-s + 1.43·31-s − 0.176·32-s + 1.39·33-s − 0.342·34-s + 1/6·36-s − 0.657·37-s + 0.640·39-s − 1.56·41-s + 0.304·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 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.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 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.ag |
| 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
−8.265197887657296296408135524905, −7.27509200434917567912613486449, −6.69230734927741041661200692585, −5.89496930434099859832664154716, −5.19767872419815523490952861147, −4.62210033843428065093542133900, −3.18140999314506805318435623178, −2.38696557728104345139988130388, −1.02088570685086890474467540436, 0,
1.02088570685086890474467540436, 2.38696557728104345139988130388, 3.18140999314506805318435623178, 4.62210033843428065093542133900, 5.19767872419815523490952861147, 5.89496930434099859832664154716, 6.69230734927741041661200692585, 7.27509200434917567912613486449, 8.265197887657296296408135524905
