| L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s + 13-s − 15-s + 2·17-s − 2·19-s − 8·23-s + 25-s + 27-s − 6·29-s + 2·31-s + 2·33-s + 2·37-s + 39-s + 2·41-s − 45-s − 6·47-s + 2·51-s − 10·53-s − 2·55-s − 2·57-s + 14·59-s + 10·61-s − 65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s + 0.312·41-s − 0.149·45-s − 0.875·47-s + 0.280·51-s − 1.37·53-s − 0.269·55-s − 0.264·57-s + 1.82·59-s + 1.28·61-s − 0.124·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.93209551850218, −12.51826329664613, −11.92857999966751, −11.42166931177347, −11.35537198947795, −10.49410089712335, −10.05566454187490, −9.771219010620955, −9.172342891956876, −8.669746997441777, −8.339978123204164, −7.736320814102849, −7.547045744765485, −6.834341425189197, −6.348487217073908, −5.943416266364760, −5.328754279175538, −4.700887614514729, −4.089391155907023, −3.815025915872462, −3.338018967097474, −2.648040108573544, −2.023131312634357, −1.559629518248588, −0.7911439437371840, 0,
0.7911439437371840, 1.559629518248588, 2.023131312634357, 2.648040108573544, 3.338018967097474, 3.815025915872462, 4.089391155907023, 4.700887614514729, 5.328754279175538, 5.943416266364760, 6.348487217073908, 6.834341425189197, 7.547045744765485, 7.736320814102849, 8.339978123204164, 8.669746997441777, 9.172342891956876, 9.771219010620955, 10.05566454187490, 10.49410089712335, 11.35537198947795, 11.42166931177347, 11.92857999966751, 12.51826329664613, 12.93209551850218
