| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s − 3·13-s − 15-s + 17-s − 5·19-s + 21-s − 7·23-s − 4·25-s + 27-s − 6·29-s − 4·31-s + 33-s − 35-s − 2·37-s − 3·39-s − 3·41-s + 7·43-s − 45-s + 4·47-s + 49-s + 51-s − 55-s − 5·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.258·15-s + 0.242·17-s − 1.14·19-s + 0.218·21-s − 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s − 0.480·39-s − 0.468·41-s + 1.06·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.140·51-s − 0.134·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−8.327962755523184219820362608650, −7.70732686237876575774406809388, −7.13488506783359951593528172749, −6.11380301370410518982512512718, −5.27865595149170458917012993184, −4.16036918305284879258260652404, −3.81627007630246057783038496567, −2.49554752778291095006682600923, −1.74167488209179237546729397318, 0,
1.74167488209179237546729397318, 2.49554752778291095006682600923, 3.81627007630246057783038496567, 4.16036918305284879258260652404, 5.27865595149170458917012993184, 6.11380301370410518982512512718, 7.13488506783359951593528172749, 7.70732686237876575774406809388, 8.327962755523184219820362608650
