| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 2·11-s − 4·14-s + 16-s + 6·17-s + 4·19-s + 2·22-s − 4·28-s − 2·29-s − 8·31-s + 32-s + 6·34-s − 2·37-s + 4·38-s − 6·41-s + 4·43-s + 2·44-s + 9·49-s + 10·53-s − 4·56-s − 2·58-s + 10·61-s − 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.603·11-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.426·22-s − 0.755·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.301·44-s + 9/7·49-s + 1.37·53-s − 0.534·56-s − 0.262·58-s + 1.28·61-s − 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.212710086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.212710086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.74315554469896, −14.41048264954248, −13.90965929082246, −13.23200478568226, −12.93951239561580, −12.30830879463403, −11.96096076438921, −11.42473721376115, −10.67533047234235, −10.09724858544060, −9.654879449690719, −9.218196503192203, −8.491057454183834, −7.704619439918530, −7.056516847809123, −6.846700761487920, −5.944891021401264, −5.623015443968300, −5.074169886781551, −4.019323876858974, −3.574826305947968, −3.213762305801041, −2.415722999239481, −1.477876304029013, −0.6057763105703913,
0.6057763105703913, 1.477876304029013, 2.415722999239481, 3.213762305801041, 3.574826305947968, 4.019323876858974, 5.074169886781551, 5.623015443968300, 5.944891021401264, 6.846700761487920, 7.056516847809123, 7.704619439918530, 8.491057454183834, 9.218196503192203, 9.654879449690719, 10.09724858544060, 10.67533047234235, 11.42473721376115, 11.96096076438921, 12.30830879463403, 12.93951239561580, 13.23200478568226, 13.90965929082246, 14.41048264954248, 14.74315554469896
