| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 14-s + 16-s − 6·17-s + 5·19-s − 20-s + 22-s + 6·23-s + 25-s + 28-s − 4·29-s + 31-s + 32-s − 6·34-s − 35-s − 8·37-s + 5·38-s − 40-s − 10·41-s + 9·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.14·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.188·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s − 1.31·37-s + 0.811·38-s − 0.158·40-s − 1.56·41-s + 1.37·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.45919383420228, −13.02223569005493, −12.60566648434855, −11.95804939085764, −11.64661026095987, −11.19597269540959, −10.84066578072801, −10.29779353305844, −9.663059819355764, −8.934328865140163, −8.871063180148054, −8.089415658287517, −7.515574785425997, −7.153534264588186, −6.614461401782654, −6.228215266541510, −5.310727327892256, −5.107658314707002, −4.601544859387450, −3.919847900646856, −3.501357059319409, −2.931593593910408, −2.244487255049948, −1.626931761652275, −0.9397010048318710, 0,
0.9397010048318710, 1.626931761652275, 2.244487255049948, 2.931593593910408, 3.501357059319409, 3.919847900646856, 4.601544859387450, 5.107658314707002, 5.310727327892256, 6.228215266541510, 6.614461401782654, 7.153534264588186, 7.515574785425997, 8.089415658287517, 8.871063180148054, 8.934328865140163, 9.663059819355764, 10.29779353305844, 10.84066578072801, 11.19597269540959, 11.64661026095987, 11.95804939085764, 12.60566648434855, 13.02223569005493, 13.45919383420228
