| L(s) = 1 | − 2·3-s − 5-s + 9-s + 2·11-s + 4·13-s + 2·15-s + 2·17-s − 4·19-s − 23-s + 25-s + 4·27-s − 2·29-s − 4·31-s − 4·33-s − 8·37-s − 8·39-s + 2·41-s − 10·43-s − 45-s − 12·47-s − 4·51-s − 4·53-s − 2·55-s + 8·57-s + 10·59-s + 14·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.31·37-s − 1.28·39-s + 0.312·41-s − 1.52·43-s − 0.149·45-s − 1.75·47-s − 0.560·51-s − 0.549·53-s − 0.269·55-s + 1.05·57-s + 1.30·59-s + 1.79·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.31317029949310, −13.38171220885151, −13.12061098048695, −12.55011954127821, −11.92252046513723, −11.73109030604971, −11.13224685908775, −10.85542827807668, −10.27417846103473, −9.784091725925119, −9.050011411337063, −8.499113593423839, −8.230743835487832, −7.448048735462349, −6.774374073762315, −6.430662916322563, −6.041066030713568, −5.181911250873185, −5.085290473897491, −4.167281877891481, −3.627673386362244, −3.271457564768452, −2.122628303195465, −1.516187978143585, −0.7357384556454294, 0,
0.7357384556454294, 1.516187978143585, 2.122628303195465, 3.271457564768452, 3.627673386362244, 4.167281877891481, 5.085290473897491, 5.181911250873185, 6.041066030713568, 6.430662916322563, 6.774374073762315, 7.448048735462349, 8.230743835487832, 8.499113593423839, 9.050011411337063, 9.784091725925119, 10.27417846103473, 10.85542827807668, 11.13224685908775, 11.73109030604971, 11.92252046513723, 12.55011954127821, 13.12061098048695, 13.38171220885151, 14.31317029949310
