| L(s) = 1 | − 2·5-s + 2·7-s − 3·9-s + 6·11-s − 13-s + 17-s − 3·23-s − 25-s − 6·29-s + 9·31-s − 4·35-s − 37-s − 3·41-s + 3·43-s + 6·45-s − 6·47-s − 3·49-s + 6·53-s − 12·55-s + 5·59-s − 7·61-s − 6·63-s + 2·65-s + 3·67-s − 8·71-s − 6·73-s + 12·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 9-s + 1.80·11-s − 0.277·13-s + 0.242·17-s − 0.625·23-s − 1/5·25-s − 1.11·29-s + 1.61·31-s − 0.676·35-s − 0.164·37-s − 0.468·41-s + 0.457·43-s + 0.894·45-s − 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.61·55-s + 0.650·59-s − 0.896·61-s − 0.755·63-s + 0.248·65-s + 0.366·67-s − 0.949·71-s − 0.702·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375040510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375040510\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.49232839826171, −11.91687644405699, −11.75752150816066, −11.55851203988222, −11.04590141427428, −10.47858068919878, −9.885154612152117, −9.372355273903709, −9.008618495026467, −8.355421254193460, −8.145974185377764, −7.736098741129467, −6.984735600836319, −6.742992397512307, −5.998626623647209, −5.701193522368114, −5.039382163319322, −4.340441609299345, −4.175800144279182, −3.518507427784449, −3.080450739213371, −2.336324931437774, −1.663570104198381, −1.159898091463740, −0.3289354346000692,
0.3289354346000692, 1.159898091463740, 1.663570104198381, 2.336324931437774, 3.080450739213371, 3.518507427784449, 4.175800144279182, 4.340441609299345, 5.039382163319322, 5.701193522368114, 5.998626623647209, 6.742992397512307, 6.984735600836319, 7.736098741129467, 8.145974185377764, 8.355421254193460, 9.008618495026467, 9.372355273903709, 9.885154612152117, 10.47858068919878, 11.04590141427428, 11.55851203988222, 11.75752150816066, 11.91687644405699, 12.49232839826171
