| L(s) = 1 | − 4·7-s + 2·13-s + 6·17-s + 6·19-s − 23-s − 5·25-s − 6·29-s − 8·37-s + 6·41-s + 2·43-s + 8·47-s + 9·49-s + 8·53-s − 4·59-s + 4·61-s + 2·67-s + 8·71-s − 6·73-s − 12·79-s + 10·83-s − 10·89-s − 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s + 1.45·17-s + 1.37·19-s − 0.208·23-s − 25-s − 1.11·29-s − 1.31·37-s + 0.937·41-s + 0.304·43-s + 1.16·47-s + 9/7·49-s + 1.09·53-s − 0.520·59-s + 0.512·61-s + 0.244·67-s + 0.949·71-s − 0.702·73-s − 1.35·79-s + 1.09·83-s − 1.05·89-s − 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.29790102269711, −12.79420645978158, −12.32982372629282, −12.04679945127251, −11.44950282843301, −10.99558138699641, −10.26134910444189, −9.978419926481790, −9.638512461700468, −9.077457979676194, −8.735032118221763, −7.900002013564666, −7.522312493059455, −7.148054562057576, −6.543438285761228, −5.891173061750510, −5.624383964967388, −5.276450635566099, −4.245596515656773, −3.706299338499024, −3.454508880597284, −2.882180993258862, −2.229009877847918, −1.380530415214980, −0.7943106390064651, 0,
0.7943106390064651, 1.380530415214980, 2.229009877847918, 2.882180993258862, 3.454508880597284, 3.706299338499024, 4.245596515656773, 5.276450635566099, 5.624383964967388, 5.891173061750510, 6.543438285761228, 7.148054562057576, 7.522312493059455, 7.900002013564666, 8.735032118221763, 9.077457979676194, 9.638512461700468, 9.978419926481790, 10.26134910444189, 10.99558138699641, 11.44950282843301, 12.04679945127251, 12.32982372629282, 12.79420645978158, 13.29790102269711
