| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 3·11-s − 2·13-s + 2·14-s + 16-s − 3·17-s + 19-s + 3·22-s + 6·23-s − 5·25-s + 2·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s + 3·34-s − 4·37-s − 38-s − 43-s − 3·44-s − 6·46-s − 6·47-s − 3·49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.657·37-s − 0.162·38-s − 0.152·43-s − 0.452·44-s − 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4373923788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4373923788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−12.80785491804183, −12.31223041567018, −11.72944772124083, −11.41500233164351, −10.78067902332705, −10.43966521772930, −9.873780802460059, −9.727991414297664, −8.942332188328114, −8.753701589114326, −8.157850721427133, −7.583774022129153, −7.199292789766286, −6.763348320526647, −6.280994059005018, −5.696840960217789, −5.126366302839620, −4.756237492654164, −3.967393478643677, −3.363287552794764, −2.857238892036190, −2.384161715569680, −1.798513342335355, −0.9837335144205626, −0.2241132897927158,
0.2241132897927158, 0.9837335144205626, 1.798513342335355, 2.384161715569680, 2.857238892036190, 3.363287552794764, 3.967393478643677, 4.756237492654164, 5.126366302839620, 5.696840960217789, 6.280994059005018, 6.763348320526647, 7.199292789766286, 7.583774022129153, 8.157850721427133, 8.753701589114326, 8.942332188328114, 9.727991414297664, 9.873780802460059, 10.43966521772930, 10.78067902332705, 11.41500233164351, 11.72944772124083, 12.31223041567018, 12.80785491804183
