| L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s − 13-s + 2·15-s + 2·17-s − 8·23-s − 25-s + 27-s + 2·29-s − 8·31-s + 4·33-s + 6·37-s − 39-s − 6·41-s + 4·43-s + 2·45-s + 12·47-s + 2·51-s − 6·53-s + 8·55-s − 4·59-s + 2·61-s − 2·65-s − 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s + 0.280·51-s − 0.824·53-s + 1.07·55-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.94482761455816, −13.45885483630634, −12.79379207807874, −12.44466121014264, −11.83336344349876, −11.54836000156175, −10.74556964534077, −10.22136018290546, −9.883750835160965, −9.361446334365144, −8.988598579706862, −8.555175161396973, −7.660472477011590, −7.584349776160369, −6.807286366173318, −6.151988492186862, −5.919446604008922, −5.326861106887364, −4.476637855258088, −4.069346949574545, −3.527423788885852, −2.829717972147563, −2.152087920289148, −1.697077030179612, −1.106287082863818, 0,
1.106287082863818, 1.697077030179612, 2.152087920289148, 2.829717972147563, 3.527423788885852, 4.069346949574545, 4.476637855258088, 5.326861106887364, 5.919446604008922, 6.151988492186862, 6.807286366173318, 7.584349776160369, 7.660472477011590, 8.555175161396973, 8.988598579706862, 9.361446334365144, 9.883750835160965, 10.22136018290546, 10.74556964534077, 11.54836000156175, 11.83336344349876, 12.44466121014264, 12.79379207807874, 13.45885483630634, 13.94482761455816
