| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s − 2·13-s + 14-s + 16-s − 2·17-s + 4·19-s + 22-s + 4·23-s − 2·26-s + 28-s − 6·29-s + 32-s − 2·34-s + 6·37-s + 4·38-s − 10·41-s − 4·43-s + 44-s + 4·46-s + 49-s − 2·52-s − 2·53-s + 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.213·22-s + 0.834·23-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.342·34-s + 0.986·37-s + 0.648·38-s − 1.56·41-s − 0.609·43-s + 0.150·44-s + 0.589·46-s + 1/7·49-s − 0.277·52-s − 0.274·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.13007307426324, −14.71547948326151, −14.19615701974307, −13.60774420356936, −13.18087312978102, −12.71634928398572, −11.88732036197002, −11.74401875321549, −11.10922725485248, −10.58657839590848, −9.935358227718080, −9.311991818423632, −8.872744064334182, −8.033935344596786, −7.542959749569439, −7.008676003866532, −6.444678781639780, −5.729131605073232, −5.164703103780253, −4.667561764048389, −4.050432994702717, −3.259810169352239, −2.772216903813262, −1.852749125621308, −1.257771976061066, 0,
1.257771976061066, 1.852749125621308, 2.772216903813262, 3.259810169352239, 4.050432994702717, 4.667561764048389, 5.164703103780253, 5.729131605073232, 6.444678781639780, 7.008676003866532, 7.542959749569439, 8.033935344596786, 8.872744064334182, 9.311991818423632, 9.935358227718080, 10.58657839590848, 11.10922725485248, 11.74401875321549, 11.88732036197002, 12.71634928398572, 13.18087312978102, 13.60774420356936, 14.19615701974307, 14.71547948326151, 15.13007307426324
