| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 7-s − 8-s + 9-s + 4·10-s + 4·11-s − 2·12-s − 4·13-s + 14-s + 8·15-s + 16-s − 18-s − 6·19-s − 4·20-s + 2·21-s − 4·22-s + 2·24-s + 11·25-s + 4·26-s + 4·27-s − 28-s − 6·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.20·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 2.06·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.894·20-s + 0.436·21-s − 0.852·22-s + 0.408·24-s + 11/5·25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s − 1.11·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.036451484351060405988469461208, −7.24053247996392521700244035449, −6.80184620180027755716897051642, −6.06962545746684461467489474971, −5.06967275697062302961538528733, −4.19061034903827668879669972114, −3.62196634868791746035532272505, −2.34523143295550698928421786225, −0.792780311714629070067728054136, 0,
0.792780311714629070067728054136, 2.34523143295550698928421786225, 3.62196634868791746035532272505, 4.19061034903827668879669972114, 5.06967275697062302961538528733, 6.06962545746684461467489474971, 6.80184620180027755716897051642, 7.24053247996392521700244035449, 8.036451484351060405988469461208
