| L(s) = 1 | − 2·5-s − 6·13-s − 2·17-s − 25-s − 10·29-s + 2·37-s − 10·41-s − 7·49-s + 14·53-s + 10·61-s + 12·65-s − 6·73-s + 4·85-s − 10·89-s + 18·97-s − 2·101-s − 6·109-s + 14·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.66·13-s − 0.485·17-s − 1/5·25-s − 1.85·29-s + 0.328·37-s − 1.56·41-s − 49-s + 1.92·53-s + 1.28·61-s + 1.48·65-s − 0.702·73-s + 0.433·85-s − 1.05·89-s + 1.82·97-s − 0.199·101-s − 0.574·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−10.23383425889508895415100406682, −9.468761705895534248392912886381, −8.439450125780651634799093893382, −7.52766744709985624839545780760, −6.93944627141196287077129229697, −5.54216083419874931368270995905, −4.56897038264081606692544495668, −3.55808048734848453068190211023, −2.19360926670427462073525005533, 0,
2.19360926670427462073525005533, 3.55808048734848453068190211023, 4.56897038264081606692544495668, 5.54216083419874931368270995905, 6.93944627141196287077129229697, 7.52766744709985624839545780760, 8.439450125780651634799093893382, 9.468761705895534248392912886381, 10.23383425889508895415100406682
