| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 6·11-s − 12-s − 2·13-s + 14-s + 16-s − 6·17-s + 18-s − 21-s + 6·22-s + 23-s − 24-s − 5·25-s − 2·26-s − 27-s + 28-s + 6·29-s − 8·31-s + 32-s − 6·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.218·21-s + 1.27·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.04·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.71620849798359, −12.17573466763581, −11.96625810533349, −11.38249476929670, −11.27384857229584, −10.65672036180699, −10.11541571256490, −9.714975934515406, −9.024371603455341, −8.784125263382987, −8.236072191700119, −7.470916299946809, −7.030421220529667, −6.669301290213011, −6.390063063916570, −5.680807674742855, −5.250198448125307, −4.776267402879175, −4.268897060457913, −3.734734455007229, −3.554678328946221, −2.443003064121195, −2.088392180450595, −1.519549511606236, −0.8452288909584823, 0,
0.8452288909584823, 1.519549511606236, 2.088392180450595, 2.443003064121195, 3.554678328946221, 3.734734455007229, 4.268897060457913, 4.776267402879175, 5.250198448125307, 5.680807674742855, 6.390063063916570, 6.669301290213011, 7.030421220529667, 7.470916299946809, 8.236072191700119, 8.784125263382987, 9.024371603455341, 9.714975934515406, 10.11541571256490, 10.65672036180699, 11.27384857229584, 11.38249476929670, 11.96625810533349, 12.17573466763581, 12.71620849798359
