| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s + 7-s − 3·8-s + 9-s + 6·11-s − 2·12-s + 13-s + 14-s − 16-s + 2·17-s + 18-s + 2·21-s + 6·22-s − 6·24-s + 26-s − 4·27-s − 28-s − 2·29-s + 8·31-s + 5·32-s + 12·33-s + 2·34-s − 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.436·21-s + 1.27·22-s − 1.22·24-s + 0.196·26-s − 0.769·27-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.883·32-s + 2.08·33-s + 0.342·34-s − 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.638830094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.638830094\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−9.056434934260356398442891224957, −8.401960253544204275790029584428, −7.64055409913096110067846710189, −6.55757565356518139890584975688, −5.84947877104722764839117549876, −4.81086156918620624513848455420, −3.93320844349636262683813018728, −3.51339207607108565108856018847, −2.45748880537956494640691448936, −1.16223872493470630872421361920,
1.16223872493470630872421361920, 2.45748880537956494640691448936, 3.51339207607108565108856018847, 3.93320844349636262683813018728, 4.81086156918620624513848455420, 5.84947877104722764839117549876, 6.55757565356518139890584975688, 7.64055409913096110067846710189, 8.401960253544204275790029584428, 9.056434934260356398442891224957
