| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 7-s + 8-s − 2·9-s + 2·10-s − 12-s + 7·13-s + 14-s − 2·15-s + 16-s − 2·18-s − 19-s + 2·20-s − 21-s + 23-s − 24-s − 25-s + 7·26-s + 5·27-s + 28-s − 8·29-s − 2·30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.288·12-s + 1.94·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.471·18-s − 0.229·19-s + 0.447·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 1.37·26-s + 0.962·27-s + 0.188·28-s − 1.48·29-s − 0.365·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.810324182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810324182\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−9.321163035338629711107560003836, −8.570437871548498622785381081464, −7.70953122368765054394545141739, −6.45531510545662697550993489967, −6.00605379979179199389429417056, −5.48606611855662502353572705359, −4.45440771892760320698174133252, −3.47012506203098234088497002359, −2.33324873277539646703587189993, −1.17556820145289401973350086061,
1.17556820145289401973350086061, 2.33324873277539646703587189993, 3.47012506203098234088497002359, 4.45440771892760320698174133252, 5.48606611855662502353572705359, 6.00605379979179199389429417056, 6.45531510545662697550993489967, 7.70953122368765054394545141739, 8.570437871548498622785381081464, 9.321163035338629711107560003836
