| L(s) = 1 | + 2·3-s − 5-s + 4·7-s + 9-s + 6·13-s − 2·15-s + 8·21-s − 8·23-s + 25-s − 4·27-s − 29-s + 8·31-s − 4·35-s + 12·39-s + 6·41-s − 2·43-s − 45-s − 2·47-s + 9·49-s − 6·53-s + 12·59-s − 6·61-s + 4·63-s − 6·65-s + 16·67-s − 16·69-s + 4·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 1.74·21-s − 1.66·23-s + 1/5·25-s − 0.769·27-s − 0.185·29-s + 1.43·31-s − 0.676·35-s + 1.92·39-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s − 0.768·61-s + 0.503·63-s − 0.744·65-s + 1.95·67-s − 1.92·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.700400016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.700400016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.616845086761185794958665415093, −8.649411087787851139834760113061, −8.114815467177020585328784763866, −7.87625769670227615420747047862, −6.51675027471048394531611568400, −5.49096139552585325936940714051, −4.29321580578426937763178235894, −3.68307562545635005531036992266, −2.43967867174375613914491274916, −1.37434495232063328769147880527,
1.37434495232063328769147880527, 2.43967867174375613914491274916, 3.68307562545635005531036992266, 4.29321580578426937763178235894, 5.49096139552585325936940714051, 6.51675027471048394531611568400, 7.87625769670227615420747047862, 8.114815467177020585328784763866, 8.649411087787851139834760113061, 9.616845086761185794958665415093
