| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 5·11-s + 2·12-s + 7·13-s − 14-s + 16-s − 18-s − 7·19-s + 2·21-s − 5·22-s − 23-s − 2·24-s − 7·26-s − 4·27-s + 28-s + 5·29-s − 10·31-s − 32-s + 10·33-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 − 0.353·8-s + 1/3·9-s + 1.50·11-s + 0.577·12-s + 1.94·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 1.60·19-s + 0.436·21-s − 1.06·22-s − 0.208·23-s − 0.408·24-s − 1.37·26-s − 0.769·27-s + 0.188·28-s + 0.928·29-s − 1.79·31-s − 0.176·32-s + 1.74·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028865657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028865657\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−9.339338249605910733656123522861, −8.921062385336984286171919459130, −8.438824827418498639061216142407, −7.63634118130800779031578111015, −6.53594006734379987075963155587, −5.92897825368943789059968061915, −4.14157188173377372653812160569, −3.59395050947079775635382559392, −2.26025502781996897382794189856, −1.28646998934505136843688398432,
1.28646998934505136843688398432, 2.26025502781996897382794189856, 3.59395050947079775635382559392, 4.14157188173377372653812160569, 5.92897825368943789059968061915, 6.53594006734379987075963155587, 7.63634118130800779031578111015, 8.438824827418498639061216142407, 8.921062385336984286171919459130, 9.339338249605910733656123522861
