| L(s) = 1 | + 7-s + 3·13-s + 17-s + 5·19-s + 4·23-s − 9·29-s + 6·31-s + 11·37-s − 2·41-s + 2·43-s − 6·47-s − 6·49-s + 4·53-s − 4·59-s − 4·61-s + 6·67-s − 5·71-s + 6·73-s + 14·79-s − 7·83-s − 4·89-s + 3·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.832·13-s + 0.242·17-s + 1.14·19-s + 0.834·23-s − 1.67·29-s + 1.07·31-s + 1.80·37-s − 0.312·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s + 0.549·53-s − 0.520·59-s − 0.512·61-s + 0.733·67-s − 0.593·71-s + 0.702·73-s + 1.57·79-s − 0.768·83-s − 0.423·89-s + 0.314·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 217800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 217800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.649584086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.649584086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−13.09660049829047, −12.57468051079492, −11.96949229970847, −11.39628400842565, −11.27987865247413, −10.80248060157694, −10.07445455810902, −9.724388004822934, −9.215300867272572, −8.813521011490672, −8.156912293624482, −7.754790297392863, −7.426278661429182, −6.707127870546151, −6.251009823633022, −5.729677586917258, −5.240134797134808, −4.700310848733082, −4.206810694458896, −3.420733899125603, −3.212164219644999, −2.421955400642261, −1.742718340400973, −1.122969554724411, −0.5955693208807949,
0.5955693208807949, 1.122969554724411, 1.742718340400973, 2.421955400642261, 3.212164219644999, 3.420733899125603, 4.206810694458896, 4.700310848733082, 5.240134797134808, 5.729677586917258, 6.251009823633022, 6.707127870546151, 7.426278661429182, 7.754790297392863, 8.156912293624482, 8.813521011490672, 9.215300867272572, 9.724388004822934, 10.07445455810902, 10.80248060157694, 11.27987865247413, 11.39628400842565, 11.96949229970847, 12.57468051079492, 13.09660049829047
