| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 13-s + 16-s + 3·17-s − 18-s + 2·19-s + 3·23-s − 24-s − 26-s + 27-s + 6·29-s + 5·31-s − 32-s − 3·34-s + 36-s − 4·37-s − 2·38-s + 39-s + 9·41-s + 8·43-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.625·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s − 0.657·37-s − 0.324·38-s + 0.160·39-s + 1.40·41-s + 1.21·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.212855710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.212855710\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.99629905160750, −13.31836607858809, −12.72293756200523, −12.40094911474597, −11.74889256933202, −11.31966876063932, −10.74150157328306, −10.19259523554920, −9.861892259811164, −9.277306298309526, −8.811764826492779, −8.349258319722053, −7.852340195365586, −7.373915601468421, −6.852086362316630, −6.302425830840580, −5.658124905566646, −5.116827247953787, −4.349864103464712, −3.803770633228984, −3.038020086396768, −2.690132004711792, −1.960244153907117, −1.071730957495225, −0.7377610801043479,
0.7377610801043479, 1.071730957495225, 1.960244153907117, 2.690132004711792, 3.038020086396768, 3.803770633228984, 4.349864103464712, 5.116827247953787, 5.658124905566646, 6.302425830840580, 6.852086362316630, 7.373915601468421, 7.852340195365586, 8.349258319722053, 8.811764826492779, 9.277306298309526, 9.861892259811164, 10.19259523554920, 10.74150157328306, 11.31966876063932, 11.74889256933202, 12.40094911474597, 12.72293756200523, 13.31836607858809, 13.99629905160750
