| L(s) = 1 | − 3-s + 9-s + 6·11-s − 13-s − 6·19-s + 6·23-s − 27-s + 2·29-s − 4·31-s − 6·33-s − 10·37-s + 39-s − 6·41-s + 8·43-s + 8·47-s − 7·49-s + 6·53-s + 6·57-s − 10·59-s − 6·61-s + 4·67-s − 6·69-s + 8·71-s − 6·73-s − 16·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 1.37·19-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 1.04·33-s − 1.64·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 49-s + 0.824·53-s + 0.794·57-s − 1.30·59-s − 0.768·61-s + 0.488·67-s − 0.722·69-s + 0.949·71-s − 0.702·73-s − 1.80·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.56349233200810, −15.56473462900255, −15.30026119733683, −14.51309871161322, −14.22870201190498, −13.49622231704560, −12.74709702802780, −12.34753671172267, −11.84640110379455, −11.20666706845611, −10.71185514520122, −10.18085902976716, −9.268108673411687, −9.002123061383821, −8.380615077855694, −7.394917402405872, −6.825941824631526, −6.481285992580358, −5.722247384057230, −5.019088213847977, −4.271086238803978, −3.807432846172683, −2.881926322503531, −1.821056556307855, −1.173457387116674, 0,
1.173457387116674, 1.821056556307855, 2.881926322503531, 3.807432846172683, 4.271086238803978, 5.019088213847977, 5.722247384057230, 6.481285992580358, 6.825941824631526, 7.394917402405872, 8.380615077855694, 9.002123061383821, 9.268108673411687, 10.18085902976716, 10.71185514520122, 11.20666706845611, 11.84640110379455, 12.34753671172267, 12.74709702802780, 13.49622231704560, 14.22870201190498, 14.51309871161322, 15.30026119733683, 15.56473462900255, 16.56349233200810
