| L(s) = 1 | + 4·5-s − 5·13-s − 17-s + 3·19-s − 2·23-s + 11·25-s + 8·29-s − 4·37-s + 6·41-s − 11·43-s + 11·47-s − 7·49-s + 6·53-s + 12·59-s − 2·61-s − 20·65-s − 67-s − 10·73-s + 4·79-s − 3·83-s − 4·85-s − 7·89-s + 12·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.38·13-s − 0.242·17-s + 0.688·19-s − 0.417·23-s + 11/5·25-s + 1.48·29-s − 0.657·37-s + 0.937·41-s − 1.67·43-s + 1.60·47-s − 49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 2.48·65-s − 0.122·67-s − 1.17·73-s + 0.450·79-s − 0.329·83-s − 0.433·85-s − 0.741·89-s + 1.23·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.09238962710644, −12.33605163785276, −12.18236841094922, −11.66941586896453, −10.96814329642597, −10.45929687376495, −10.06045508925741, −9.808747989308071, −9.411699780944190, −8.797175219899027, −8.493291644152983, −7.755662829977376, −7.251043683037105, −6.690535325727817, −6.468746249578206, −5.748443441847905, −5.324762605075974, −5.051899088329983, −4.433431722007150, −3.809176137452545, −2.815987390278552, −2.723124328678898, −2.111977759507907, −1.507663798852493, −0.9395978595679754, 0,
0.9395978595679754, 1.507663798852493, 2.111977759507907, 2.723124328678898, 2.815987390278552, 3.809176137452545, 4.433431722007150, 5.051899088329983, 5.324762605075974, 5.748443441847905, 6.468746249578206, 6.690535325727817, 7.251043683037105, 7.755662829977376, 8.493291644152983, 8.797175219899027, 9.411699780944190, 9.808747989308071, 10.06045508925741, 10.45929687376495, 10.96814329642597, 11.66941586896453, 12.18236841094922, 12.33605163785276, 13.09238962710644
