| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 5·11-s − 13-s + 15-s + 19-s − 2·21-s + 4·23-s + 25-s − 27-s + 5·29-s + 4·31-s − 5·33-s − 2·35-s + 6·37-s + 39-s − 41-s − 8·43-s − 45-s − 4·47-s − 3·49-s − 12·53-s − 5·55-s − 57-s + 11·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s + 0.229·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 0.718·31-s − 0.870·33-s − 0.338·35-s + 0.986·37-s + 0.160·39-s − 0.156·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s − 1.64·53-s − 0.674·55-s − 0.132·57-s + 1.43·59-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.12643635857120, −12.62637649717796, −12.07166689644908, −11.68342426770576, −11.44971399929573, −11.10066887894673, −10.34227256809793, −10.02401382007608, −9.440971681642931, −8.937377949474509, −8.469174591126927, −7.990796789340766, −7.476028415495947, −6.859612011191711, −6.590698185032306, −6.050924353270321, −5.425173578184468, −4.809482045691784, −4.450368721162571, −4.120079142914067, −3.174129860308422, −2.960373276278287, −1.874830347489888, −1.371860655285939, −0.9115579491780955, 0,
0.9115579491780955, 1.371860655285939, 1.874830347489888, 2.960373276278287, 3.174129860308422, 4.120079142914067, 4.450368721162571, 4.809482045691784, 5.425173578184468, 6.050924353270321, 6.590698185032306, 6.859612011191711, 7.476028415495947, 7.990796789340766, 8.469174591126927, 8.937377949474509, 9.440971681642931, 10.02401382007608, 10.34227256809793, 11.10066887894673, 11.44971399929573, 11.68342426770576, 12.07166689644908, 12.62637649717796, 13.12643635857120
