| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·13-s + 2·15-s − 4·17-s + 4·19-s − 6·23-s − 25-s − 27-s + 2·29-s + 2·31-s + 2·37-s + 2·39-s − 6·43-s − 2·45-s + 12·47-s + 4·51-s − 4·57-s + 12·59-s + 2·61-s + 4·65-s + 4·67-s + 6·69-s − 6·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s + 0.917·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.328·37-s + 0.320·39-s − 0.914·43-s − 0.298·45-s + 1.75·47-s + 0.560·51-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.722·69-s − 0.712·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.59266244486135, −13.12558918497898, −12.56836080212151, −12.01242565870503, −11.71720566186855, −11.46454405424459, −10.83346704187273, −10.21456514648210, −9.932170719456858, −9.379877587937633, −8.573042102975590, −8.406695531634558, −7.585627862696765, −7.326273968473079, −6.854158371884638, −6.122521401982794, −5.762364939958603, −5.080170354695411, −4.555595313061634, −4.068549849442699, −3.645334649519116, −2.795623001148168, −2.270899109110398, −1.469357243359997, −0.6366507680496054, 0,
0.6366507680496054, 1.469357243359997, 2.270899109110398, 2.795623001148168, 3.645334649519116, 4.068549849442699, 4.555595313061634, 5.080170354695411, 5.762364939958603, 6.122521401982794, 6.854158371884638, 7.326273968473079, 7.585627862696765, 8.406695531634558, 8.573042102975590, 9.379877587937633, 9.932170719456858, 10.21456514648210, 10.83346704187273, 11.46454405424459, 11.71720566186855, 12.01242565870503, 12.56836080212151, 13.12558918497898, 13.59266244486135
