| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 11-s − 13-s − 15-s + 19-s − 2·21-s + 23-s + 25-s − 27-s − 29-s − 11·31-s + 33-s + 2·35-s + 3·37-s + 39-s + 2·41-s + 43-s + 45-s + 2·47-s − 3·49-s + 6·53-s − 55-s − 57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.229·19-s − 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 1.97·31-s + 0.174·33-s + 0.338·35-s + 0.493·37-s + 0.160·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.134·55-s − 0.132·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330439171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330439171\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.87256164272092, −12.70427468054845, −11.77523642309265, −11.63364871558040, −11.08921874035498, −10.69367344972301, −10.22740637939697, −9.681844953337194, −9.319647621823472, −8.665413786800395, −8.301132945003577, −7.609484293965651, −7.198880438442945, −6.889730273586107, −6.032373946897017, −5.713674781702416, −5.232250359946149, −4.840891218687639, −4.204740486403766, −3.677601888173210, −2.989929938905520, −2.197923512632126, −1.897764495901657, −1.104514067237730, −0.4739257823892758,
0.4739257823892758, 1.104514067237730, 1.897764495901657, 2.197923512632126, 2.989929938905520, 3.677601888173210, 4.204740486403766, 4.840891218687639, 5.232250359946149, 5.713674781702416, 6.032373946897017, 6.889730273586107, 7.198880438442945, 7.609484293965651, 8.301132945003577, 8.665413786800395, 9.319647621823472, 9.681844953337194, 10.22740637939697, 10.69367344972301, 11.08921874035498, 11.63364871558040, 11.77523642309265, 12.70427468054845, 12.87256164272092
