| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 9-s + 11-s − 2·12-s + 4·13-s − 16-s + 4·17-s − 18-s − 22-s + 4·23-s + 6·24-s − 4·26-s − 4·27-s − 6·29-s − 10·31-s − 5·32-s + 2·33-s − 4·34-s − 36-s + 6·37-s + 8·39-s − 4·41-s − 12·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.213·22-s + 0.834·23-s + 1.22·24-s − 0.784·26-s − 0.769·27-s − 1.11·29-s − 1.79·31-s − 0.883·32-s + 0.348·33-s − 0.685·34-s − 1/6·36-s + 0.986·37-s + 1.28·39-s − 0.624·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.60295708968728, −16.14211597423526, −15.09272065513135, −14.73671119788841, −14.43639734525800, −13.47698272039105, −13.34396563764280, −12.87291924544789, −11.83134908121249, −11.26540035462002, −10.61696881428753, −9.970083287847645, −9.330537476257915, −8.999675687038292, −8.484982635216026, −7.880135273392418, −7.464012930378308, −6.652709766781840, −5.694304880205230, −5.132760439000540, −4.116484092664262, −3.558324709968987, −3.049775154285709, −1.793431495010809, −1.329470434383808, 0,
1.329470434383808, 1.793431495010809, 3.049775154285709, 3.558324709968987, 4.116484092664262, 5.132760439000540, 5.694304880205230, 6.652709766781840, 7.464012930378308, 7.880135273392418, 8.484982635216026, 8.999675687038292, 9.330537476257915, 9.970083287847645, 10.61696881428753, 11.26540035462002, 11.83134908121249, 12.87291924544789, 13.34396563764280, 13.47698272039105, 14.43639734525800, 14.73671119788841, 15.09272065513135, 16.14211597423526, 16.60295708968728
