| L(s) = 1 | + 2·7-s + 11-s − 17-s + 2·19-s + 2·23-s − 10·29-s + 8·31-s + 6·37-s − 2·41-s − 2·43-s − 3·49-s − 12·53-s − 10·59-s + 10·61-s + 4·67-s − 6·71-s + 2·77-s − 10·79-s − 4·83-s + 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 0.242·17-s + 0.458·19-s + 0.417·23-s − 1.85·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.304·43-s − 3/7·49-s − 1.64·53-s − 1.30·59-s + 1.28·61-s + 0.488·67-s − 0.712·71-s + 0.227·77-s − 1.12·79-s − 0.439·83-s + 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.84474859416712, −12.29842503986774, −11.84132566481739, −11.31718082292550, −11.18624587428728, −10.65656807965656, −9.994032089213889, −9.578934876230047, −9.278623915649089, −8.561362404118397, −8.296393082695568, −7.693615252382850, −7.372243306598685, −6.815502203217080, −6.208674634179963, −5.862816419599139, −5.240665068936117, −4.608405626998259, −4.531201308317647, −3.644906630828560, −3.264544341644602, −2.600107560057033, −1.950496284145432, −1.465446559839216, −0.8451469625936646, 0,
0.8451469625936646, 1.465446559839216, 1.950496284145432, 2.600107560057033, 3.264544341644602, 3.644906630828560, 4.531201308317647, 4.608405626998259, 5.240665068936117, 5.862816419599139, 6.208674634179963, 6.815502203217080, 7.372243306598685, 7.693615252382850, 8.296393082695568, 8.561362404118397, 9.278623915649089, 9.578934876230047, 9.994032089213889, 10.65656807965656, 11.18624587428728, 11.31718082292550, 11.84132566481739, 12.29842503986774, 12.84474859416712
