| L(s) = 1 | + 7-s − 5·13-s + 6·19-s − 23-s − 5·25-s − 31-s − 9·37-s + 7·41-s + 10·43-s − 6·47-s − 6·49-s + 4·53-s + 59-s + 10·61-s + 15·67-s − 2·71-s − 2·73-s − 12·79-s + 9·83-s − 10·89-s − 5·91-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.38·13-s + 1.37·19-s − 0.208·23-s − 25-s − 0.179·31-s − 1.47·37-s + 1.09·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s + 0.549·53-s + 0.130·59-s + 1.28·61-s + 1.83·67-s − 0.237·71-s − 0.234·73-s − 1.35·79-s + 0.987·83-s − 1.05·89-s − 0.524·91-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.25287466663669, −12.78399335688494, −12.31158551268770, −11.85744852700165, −11.53689380811053, −11.00370223813953, −10.45744311613008, −9.889732672579500, −9.549051646729066, −9.249484055441367, −8.403890954667796, −8.079864707547088, −7.498108463661989, −7.159171178146412, −6.713646562736476, −5.869692573800041, −5.476469342369568, −5.079709399354747, −4.486836088709626, −3.910532693268660, −3.363234673478634, −2.658806754009322, −2.196568601509477, −1.545646047648602, −0.7833420230766243, 0,
0.7833420230766243, 1.545646047648602, 2.196568601509477, 2.658806754009322, 3.363234673478634, 3.910532693268660, 4.486836088709626, 5.079709399354747, 5.476469342369568, 5.869692573800041, 6.713646562736476, 7.159171178146412, 7.498108463661989, 8.079864707547088, 8.403890954667796, 9.249484055441367, 9.549051646729066, 9.889732672579500, 10.45744311613008, 11.00370223813953, 11.53689380811053, 11.85744852700165, 12.31158551268770, 12.78399335688494, 13.25287466663669
