| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 14-s + 16-s + 4·17-s + 6·19-s + 20-s + 25-s + 28-s − 6·29-s − 4·31-s − 32-s − 4·34-s + 35-s + 8·37-s − 6·38-s − 40-s + 2·41-s + 6·43-s + 10·47-s + 49-s − 50-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.223·20-s + 1/5·25-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s + 1.31·37-s − 0.973·38-s − 0.158·40-s + 0.312·41-s + 0.914·43-s + 1.45·47-s + 1/7·49-s − 0.141·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.11485590611913, −13.41832353492056, −12.97108183255464, −12.32828687702825, −11.98245915146284, −11.36074017540589, −11.00241728242636, −10.43812747016893, −9.978425851028266, −9.314882105997921, −9.237972494350761, −8.585282552027669, −7.772603073448679, −7.468532113159578, −7.307538366375008, −6.250613213101571, −5.841838628940067, −5.504153115750012, −4.767343573868560, −4.122284426474365, −3.360550868753570, −2.860847360486334, −2.192458099135472, −1.386494152626923, −1.057199068528733, 0,
1.057199068528733, 1.386494152626923, 2.192458099135472, 2.860847360486334, 3.360550868753570, 4.122284426474365, 4.767343573868560, 5.504153115750012, 5.841838628940067, 6.250613213101571, 7.307538366375008, 7.468532113159578, 7.772603073448679, 8.585282552027669, 9.237972494350761, 9.314882105997921, 9.978425851028266, 10.43812747016893, 11.00241728242636, 11.36074017540589, 11.98245915146284, 12.32828687702825, 12.97108183255464, 13.41832353492056, 14.11485590611913
