| L(s) = 1 | + 3-s + 7-s + 9-s − 5·11-s − 13-s + 17-s + 3·19-s + 21-s + 5·23-s + 27-s − 2·29-s − 5·33-s − 2·37-s − 39-s + 7·41-s − 3·43-s − 12·47-s + 49-s + 51-s − 8·53-s + 3·57-s − 8·59-s + 6·61-s + 63-s + 8·67-s + 5·69-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.242·17-s + 0.688·19-s + 0.218·21-s + 1.04·23-s + 0.192·27-s − 0.371·29-s − 0.870·33-s − 0.328·37-s − 0.160·39-s + 1.09·41-s − 0.457·43-s − 1.75·47-s + 1/7·49-s + 0.140·51-s − 1.09·53-s + 0.397·57-s − 1.04·59-s + 0.768·61-s + 0.125·63-s + 0.977·67-s + 0.601·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.21164927384967, −14.63550013784045, −14.17256231833901, −13.63749781317255, −13.04899636222628, −12.73677190007756, −12.16320845191700, −11.28954746890745, −11.05523820618692, −10.38506898682981, −9.758632764423103, −9.436528123260235, −8.645201125889379, −8.124856581276410, −7.684357544021706, −7.233460546035777, −6.524646940410374, −5.721899091932174, −4.957098522442593, −4.912512900220101, −3.833775269680686, −3.144519244234782, −2.674829045118261, −1.923362235318969, −1.092529388804019, 0,
1.092529388804019, 1.923362235318969, 2.674829045118261, 3.144519244234782, 3.833775269680686, 4.912512900220101, 4.957098522442593, 5.721899091932174, 6.524646940410374, 7.233460546035777, 7.684357544021706, 8.124856581276410, 8.645201125889379, 9.436528123260235, 9.758632764423103, 10.38506898682981, 11.05523820618692, 11.28954746890745, 12.16320845191700, 12.73677190007756, 13.04899636222628, 13.63749781317255, 14.17256231833901, 14.63550013784045, 15.21164927384967
