| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 6·11-s + 12-s + 14-s + 16-s + 2·17-s − 18-s + 4·19-s − 21-s + 6·22-s + 6·23-s − 24-s + 27-s − 28-s + 2·29-s − 4·31-s − 32-s − 6·33-s − 2·34-s + 36-s − 6·37-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 + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 1.27·22-s + 1.25·23-s − 0.204·24-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s − 1.04·33-s − 0.342·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.33484983723614, −13.01852451671439, −12.40209635228338, −12.15458334654748, −11.37686203504879, −10.85256126897400, −10.51238480746258, −10.07939325802599, −9.614301528399032, −9.058664165001115, −8.711722492135630, −8.161089834495482, −7.592691767947854, −7.303571527259928, −6.965358707430057, −6.093419139076685, −5.516478113469160, −5.222340238608613, −4.538808675827476, −3.688702943297133, −3.240589562081073, −2.589164499590989, −2.416432621022968, −1.410182868354167, −0.8166573540359391, 0,
0.8166573540359391, 1.410182868354167, 2.416432621022968, 2.589164499590989, 3.240589562081073, 3.688702943297133, 4.538808675827476, 5.222340238608613, 5.516478113469160, 6.093419139076685, 6.965358707430057, 7.303571527259928, 7.592691767947854, 8.161089834495482, 8.711722492135630, 9.058664165001115, 9.614301528399032, 10.07939325802599, 10.51238480746258, 10.85256126897400, 11.37686203504879, 12.15458334654748, 12.40209635228338, 13.01852451671439, 13.33484983723614
