| L(s) = 1 | + 3-s + 5-s + 9-s + 13-s + 15-s − 6·17-s + 5·19-s − 6·23-s + 25-s + 27-s − 6·29-s + 5·31-s − 7·37-s + 39-s − 12·41-s + 43-s + 45-s + 6·47-s − 6·51-s + 5·57-s − 6·59-s − 2·61-s + 65-s + 7·67-s − 6·69-s − 12·71-s − 11·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 1.45·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.898·31-s − 1.15·37-s + 0.160·39-s − 1.87·41-s + 0.152·43-s + 0.149·45-s + 0.875·47-s − 0.840·51-s + 0.662·57-s − 0.781·59-s − 0.256·61-s + 0.124·65-s + 0.855·67-s − 0.722·69-s − 1.42·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.68614605088920, −15.91374008401061, −15.59270305620372, −15.07085007342480, −14.26752132313405, −13.75725780481759, −13.49388850832844, −12.86505367055873, −12.01760960084310, −11.65468001839493, −10.80777152804652, −10.25419888023252, −9.681516963528658, −9.038198566725190, −8.583268843577876, −7.885875281471013, −7.181477459433337, −6.600478596596501, −5.873051395654711, −5.178508372051537, −4.385405618472048, −3.689901881105421, −2.921023060604828, −2.085353079236348, −1.437406165058516, 0,
1.437406165058516, 2.085353079236348, 2.921023060604828, 3.689901881105421, 4.385405618472048, 5.178508372051537, 5.873051395654711, 6.600478596596501, 7.181477459433337, 7.885875281471013, 8.583268843577876, 9.038198566725190, 9.681516963528658, 10.25419888023252, 10.80777152804652, 11.65468001839493, 12.01760960084310, 12.86505367055873, 13.49388850832844, 13.75725780481759, 14.26752132313405, 15.07085007342480, 15.59270305620372, 15.91374008401061, 16.68614605088920
