| L(s) = 1 | − 2·5-s − 2·7-s − 13-s + 17-s + 19-s + 8·23-s − 25-s + 6·29-s + 4·31-s + 4·35-s + 2·37-s + 4·41-s + 7·43-s + 13·47-s − 3·49-s + 6·53-s − 4·59-s + 2·65-s − 3·67-s + 12·71-s − 8·73-s − 14·79-s − 17·83-s − 2·85-s + 17·89-s + 2·91-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.277·13-s + 0.242·17-s + 0.229·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s + 0.624·41-s + 1.06·43-s + 1.89·47-s − 3/7·49-s + 0.824·53-s − 0.520·59-s + 0.248·65-s − 0.366·67-s + 1.42·71-s − 0.936·73-s − 1.57·79-s − 1.86·83-s − 0.216·85-s + 1.80·89-s + 0.209·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.84076516102023, −12.50026183023044, −12.02167075260010, −11.52371820431460, −11.27002978371302, −10.53234633274062, −10.26454807652393, −9.754750837298734, −9.105470439089415, −8.878980470056755, −8.336514669121476, −7.626507550408439, −7.417463617584839, −6.981261793826289, −6.321939868819752, −5.943267578422514, −5.353176545624526, −4.622880809909100, −4.427369820204306, −3.686070868596787, −3.264672343184872, −2.698133838302892, −2.305988929158164, −1.113099286274031, −0.8379496867395989, 0,
0.8379496867395989, 1.113099286274031, 2.305988929158164, 2.698133838302892, 3.264672343184872, 3.686070868596787, 4.427369820204306, 4.622880809909100, 5.353176545624526, 5.943267578422514, 6.321939868819752, 6.981261793826289, 7.417463617584839, 7.626507550408439, 8.336514669121476, 8.878980470056755, 9.105470439089415, 9.754750837298734, 10.26454807652393, 10.53234633274062, 11.27002978371302, 11.52371820431460, 12.02167075260010, 12.50026183023044, 12.84076516102023
