| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 5·11-s + 13-s + 2·14-s + 16-s − 19-s + 20-s + 5·22-s − 4·23-s + 25-s + 26-s + 2·28-s − 4·29-s − 9·31-s + 32-s + 2·35-s + 9·37-s − 38-s + 40-s + 43-s + 5·44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s − 0.742·29-s − 1.61·31-s + 0.176·32-s + 0.338·35-s + 1.47·37-s − 0.162·38-s + 0.158·40-s + 0.152·43-s + 0.753·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.77665326747758, −12.41579962654848, −11.86518372265447, −11.44179874902571, −11.19067106043368, −10.67902687636681, −10.14634354657963, −9.526538655414294, −9.250553928443839, −8.667657404835852, −8.264772946032164, −7.573196625796253, −7.218874780661517, −6.739126985902791, −6.039602269282220, −5.862276284022243, −5.406511153997810, −4.641654698267861, −4.213331254020838, −3.915897315760459, −3.304320211179127, −2.599935887305654, −1.992876653017067, −1.525206403623601, −1.093768581484090, 0,
1.093768581484090, 1.525206403623601, 1.992876653017067, 2.599935887305654, 3.304320211179127, 3.915897315760459, 4.213331254020838, 4.641654698267861, 5.406511153997810, 5.862276284022243, 6.039602269282220, 6.739126985902791, 7.218874780661517, 7.573196625796253, 8.264772946032164, 8.667657404835852, 9.250553928443839, 9.526538655414294, 10.14634354657963, 10.67902687636681, 11.19067106043368, 11.44179874902571, 11.86518372265447, 12.41579962654848, 12.77665326747758
