| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 13-s + 2·15-s + 4·17-s − 21-s − 6·23-s − 25-s − 27-s − 6·29-s + 8·31-s − 2·35-s − 2·37-s + 39-s − 10·41-s − 12·43-s − 2·45-s + 6·47-s + 49-s − 4·51-s − 14·53-s − 10·59-s + 14·61-s + 63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.970·17-s − 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.160·39-s − 1.56·41-s − 1.82·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s − 1.92·53-s − 1.30·59-s + 1.79·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−9.666479737432022341657876652436, −8.353611220897493147635311982733, −7.86639633472659335661595547800, −7.00698323553803147463823568462, −6.00710715342184608077654518797, −5.09954953433320916086963398354, −4.21235264055522164803546340431, −3.27145546350264941943559110124, −1.65729275025151815856752182707, 0,
1.65729275025151815856752182707, 3.27145546350264941943559110124, 4.21235264055522164803546340431, 5.09954953433320916086963398354, 6.00710715342184608077654518797, 7.00698323553803147463823568462, 7.86639633472659335661595547800, 8.353611220897493147635311982733, 9.666479737432022341657876652436
