| L(s) = 1 | + 5·7-s + 7·13-s + 19-s − 5·25-s − 7·31-s − 10·37-s − 5·43-s + 18·49-s + 14·61-s + 16·67-s − 17·73-s − 17·79-s + 35·91-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.94·13-s + 0.229·19-s − 25-s − 1.25·31-s − 1.64·37-s − 0.762·43-s + 18/7·49-s + 1.79·61-s + 1.95·67-s − 1.98·73-s − 1.91·79-s + 3.66·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 83 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.30501116187932, −12.65580307655815, −11.95404625654380, −11.60405210758258, −11.27294564707465, −10.93357542146055, −10.37236640594349, −10.02574153371753, −9.152592584349796, −8.776078875131065, −8.438291609370867, −8.000533380504447, −7.569881782684090, −6.956856347799126, −6.499133858613212, −5.707819288805468, −5.438471966657674, −5.061362901845978, −4.245795981000433, −3.870492900746607, −3.496456834394401, −2.590680966227064, −1.859411629373033, −1.526687497062629, −1.044164798797656, 0,
1.044164798797656, 1.526687497062629, 1.859411629373033, 2.590680966227064, 3.496456834394401, 3.870492900746607, 4.245795981000433, 5.061362901845978, 5.438471966657674, 5.707819288805468, 6.499133858613212, 6.956856347799126, 7.569881782684090, 8.000533380504447, 8.438291609370867, 8.776078875131065, 9.152592584349796, 10.02574153371753, 10.37236640594349, 10.93357542146055, 11.27294564707465, 11.60405210758258, 11.95404625654380, 12.65580307655815, 13.30501116187932
