| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 5·11-s + 12-s − 4·14-s + 16-s + 2·17-s + 18-s − 2·19-s − 4·21-s − 5·22-s + 7·23-s + 24-s + 27-s − 4·28-s + 2·29-s − 2·31-s + 32-s − 5·33-s + 2·34-s + 36-s + 3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.458·19-s − 0.872·21-s − 1.06·22-s + 1.45·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.870·33-s + 0.342·34-s + 1/6·36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−15.54756207276007, −15.08872773481666, −14.62501499141475, −13.83082708498798, −13.39740172561352, −13.04846876849336, −12.44656722955808, −12.33049202594356, −11.13438462985272, −10.79382561915479, −10.14697996645582, −9.707814740846635, −9.033993191559539, −8.483988822930016, −7.629856177844427, −7.278750830360763, −6.671598157290903, −5.864793394343702, −5.553473611291798, −4.636275647959041, −4.088257558704781, −3.153972772931795, −2.904674561594679, −2.351874058400841, −1.122924990226846, 0,
1.122924990226846, 2.351874058400841, 2.904674561594679, 3.153972772931795, 4.088257558704781, 4.636275647959041, 5.553473611291798, 5.864793394343702, 6.671598157290903, 7.278750830360763, 7.629856177844427, 8.483988822930016, 9.033993191559539, 9.707814740846635, 10.14697996645582, 10.79382561915479, 11.13438462985272, 12.33049202594356, 12.44656722955808, 13.04846876849336, 13.39740172561352, 13.83082708498798, 14.62501499141475, 15.08872773481666, 15.54756207276007
