| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s + 3·17-s + 19-s − 6·23-s − 5·25-s + 2·26-s − 2·28-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 4·37-s − 38-s − 9·41-s + 43-s + 6·46-s − 6·47-s − 3·49-s + 5·50-s − 2·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 1.25·23-s − 25-s + 0.392·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s − 0.162·38-s − 1.40·41-s + 0.152·43-s + 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s − 0.277·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19602 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19602 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4875354165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4875354165\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.79900807354721, −15.27245702507352, −14.65665077566621, −14.15227321394935, −13.40916036873253, −12.96794554491544, −12.23079006965543, −11.80989382266258, −11.35835920593545, −10.39321071749452, −10.06461301092542, −9.663831716840210, −9.004254365079143, −8.406634225617645, −7.667716904891273, −7.309196104041693, −6.629967203328484, −5.834117838002178, −5.529992839855687, −4.513600454236545, −3.607254434308932, −3.219667523894214, −2.161272235621295, −1.611706732870128, −0.3117904902196710,
0.3117904902196710, 1.611706732870128, 2.161272235621295, 3.219667523894214, 3.607254434308932, 4.513600454236545, 5.529992839855687, 5.834117838002178, 6.629967203328484, 7.309196104041693, 7.667716904891273, 8.406634225617645, 9.004254365079143, 9.663831716840210, 10.06461301092542, 10.39321071749452, 11.35835920593545, 11.80989382266258, 12.23079006965543, 12.96794554491544, 13.40916036873253, 14.15227321394935, 14.65665077566621, 15.27245702507352, 15.79900807354721
