| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 12-s − 4·13-s + 2·14-s + 16-s − 6·17-s + 18-s − 19-s − 2·21-s − 24-s − 4·26-s − 27-s + 2·28-s + 8·31-s + 32-s − 6·34-s + 36-s − 8·37-s − 38-s + 4·39-s + 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s − 0.436·21-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.377·28-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.31·37-s − 0.162·38-s + 0.640·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.73258298522424, −12.30278730437152, −11.97238668167655, −11.37325876600375, −11.15089948150474, −10.69157144697711, −10.18122571976662, −9.753598876233597, −9.125939424847143, −8.664964300223689, −8.100198147538922, −7.608224791898433, −7.163251750087835, −6.624974372470258, −6.352034621831133, −5.591358298032246, −5.308771146480091, −4.662641675401265, −4.366475200071704, −4.079767971053988, −3.087689908316925, −2.632333451349661, −2.082337037673579, −1.570064778715581, −0.7584918679618581, 0,
0.7584918679618581, 1.570064778715581, 2.082337037673579, 2.632333451349661, 3.087689908316925, 4.079767971053988, 4.366475200071704, 4.662641675401265, 5.308771146480091, 5.591358298032246, 6.352034621831133, 6.624974372470258, 7.163251750087835, 7.608224791898433, 8.100198147538922, 8.664964300223689, 9.125939424847143, 9.753598876233597, 10.18122571976662, 10.69157144697711, 11.15089948150474, 11.37325876600375, 11.97238668167655, 12.30278730437152, 12.73258298522424
