| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 6·13-s + 15-s + 2·17-s + 8·19-s − 21-s + 25-s + 27-s − 6·29-s − 33-s − 35-s − 10·37-s − 6·39-s − 2·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 10·53-s − 55-s + 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.169·35-s − 1.64·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.134·55-s + 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−7.47734703871368002922563857374, −6.91423696133026922850813890603, −5.92714819929058745566748931795, −5.22346122303530198663923685987, −4.77124588459378023443590491648, −3.56723813034226821562622592153, −3.06793486714835139954663806619, −2.27138899941067196842564068584, −1.39403613941301497854176622757, 0,
1.39403613941301497854176622757, 2.27138899941067196842564068584, 3.06793486714835139954663806619, 3.56723813034226821562622592153, 4.77124588459378023443590491648, 5.22346122303530198663923685987, 5.92714819929058745566748931795, 6.91423696133026922850813890603, 7.47734703871368002922563857374
