| L(s) = 1 | + 2·5-s + 4·7-s − 3·9-s + 2·13-s + 2·17-s + 2·19-s + 6·23-s − 25-s − 6·29-s + 2·31-s + 8·35-s − 37-s − 2·41-s + 2·43-s − 6·45-s + 4·47-s + 9·49-s − 6·53-s − 6·59-s − 6·61-s − 12·63-s + 4·65-s + 8·67-s + 4·71-s + 14·73-s − 2·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 9-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 1.25·23-s − 1/5·25-s − 1.11·29-s + 0.359·31-s + 1.35·35-s − 0.164·37-s − 0.312·41-s + 0.304·43-s − 0.894·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.781·59-s − 0.768·61-s − 1.51·63-s + 0.496·65-s + 0.977·67-s + 0.474·71-s + 1.63·73-s − 0.225·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.228500393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228500393\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 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
−9.643598765820474658303483164118, −8.939908363360539694112713323446, −8.179854551220346134794927370536, −7.44643679398612564054429895074, −6.22415831207096217368749210189, −5.46349139552603568969562413832, −4.88253984327955854319240680173, −3.50211556528427241737853655370, −2.30384470940550542264449691743, −1.26629318215268350130600876569,
1.26629318215268350130600876569, 2.30384470940550542264449691743, 3.50211556528427241737853655370, 4.88253984327955854319240680173, 5.46349139552603568969562413832, 6.22415831207096217368749210189, 7.44643679398612564054429895074, 8.179854551220346134794927370536, 8.939908363360539694112713323446, 9.643598765820474658303483164118
