| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 2·9-s − 10-s + 12-s + 4·13-s + 15-s + 16-s + 8·17-s + 2·18-s − 19-s + 20-s + 5·23-s − 24-s + 25-s − 4·26-s − 5·27-s − 3·29-s − 30-s − 2·31-s − 32-s − 8·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s + 1.94·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 1.04·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.962·27-s − 0.557·29-s − 0.182·30-s − 0.359·31-s − 0.176·32-s − 1.37·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224874247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224874247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.87830612547539244038244085089, −7.24489913121926546413343207344, −6.33987359833703583046103837522, −5.75051329941539782115882290010, −5.21681239209785594417734063439, −3.91788032933759710853462308982, −3.25358463791154476192374828153, −2.62645613747516749673976398880, −1.61798690286758371100534669135, −0.814833936495207955295525018375,
0.814833936495207955295525018375, 1.61798690286758371100534669135, 2.62645613747516749673976398880, 3.25358463791154476192374828153, 3.91788032933759710853462308982, 5.21681239209785594417734063439, 5.75051329941539782115882290010, 6.33987359833703583046103837522, 7.24489913121926546413343207344, 7.87830612547539244038244085089
