| L(s) = 1 | + 2-s − 4-s + 5-s − 4·7-s − 3·8-s + 10-s − 4·11-s + 2·13-s − 4·14-s − 16-s − 6·17-s + 6·19-s − 20-s − 4·22-s + 6·23-s + 25-s + 2·26-s + 4·28-s − 2·29-s − 4·31-s + 5·32-s − 6·34-s − 4·35-s − 8·37-s + 6·38-s − 3·40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.223·20-s − 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 0.371·29-s − 0.718·31-s + 0.883·32-s − 1.02·34-s − 0.676·35-s − 1.31·37-s + 0.973·38-s − 0.474·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.93429932073368, −13.56091816737190, −13.27159744769927, −12.81793806345016, −12.60336252049475, −11.86296059407620, −11.20117405089757, −10.75833110160866, −10.11851137269091, −9.646469070688584, −9.264321593832550, −8.736316543417715, −8.326028312539359, −7.404975950654436, −6.771621928620242, −6.613854815263387, −5.701773390263374, −5.370730489611720, −5.032136039717510, −4.147139577406149, −3.602545781817830, −3.036046328806164, −2.713278513775846, −1.795101303605961, −0.6848649562073236, 0,
0.6848649562073236, 1.795101303605961, 2.713278513775846, 3.036046328806164, 3.602545781817830, 4.147139577406149, 5.032136039717510, 5.370730489611720, 5.701773390263374, 6.613854815263387, 6.771621928620242, 7.404975950654436, 8.326028312539359, 8.736316543417715, 9.264321593832550, 9.646469070688584, 10.11851137269091, 10.75833110160866, 11.20117405089757, 11.86296059407620, 12.60336252049475, 12.81793806345016, 13.27159744769927, 13.56091816737190, 13.93429932073368
