| L(s) = 1 | + 6·13-s − 17-s + 4·19-s + 4·23-s − 5·25-s + 8·31-s − 4·37-s + 6·41-s + 8·43-s − 8·47-s − 7·49-s − 10·53-s − 12·61-s − 8·67-s + 12·71-s − 2·73-s − 4·79-s − 16·83-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.66·13-s − 0.242·17-s + 0.917·19-s + 0.834·23-s − 25-s + 1.43·31-s − 0.657·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 49-s − 1.37·53-s − 1.53·61-s − 0.977·67-s + 1.42·71-s − 0.234·73-s − 0.450·79-s − 1.75·83-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−12.89899535327330, −12.54465031831879, −11.98623388637350, −11.38614703445386, −11.16292579952274, −10.81290990909137, −10.13252721102170, −9.649145077120780, −9.336166750634297, −8.685039763929958, −8.363942179643884, −7.845933728014993, −7.400698373298976, −6.778705281633151, −6.305642916398478, −5.877196196990477, −5.500283814467292, −4.622094216454769, −4.465453934766768, −3.701745705438684, −3.152763562497107, −2.881962378105493, −1.932509326780369, −1.378330846150364, −0.9138608159844345, 0,
0.9138608159844345, 1.378330846150364, 1.932509326780369, 2.881962378105493, 3.152763562497107, 3.701745705438684, 4.465453934766768, 4.622094216454769, 5.500283814467292, 5.877196196990477, 6.305642916398478, 6.778705281633151, 7.400698373298976, 7.845933728014993, 8.363942179643884, 8.685039763929958, 9.336166750634297, 9.649145077120780, 10.13252721102170, 10.81290990909137, 11.16292579952274, 11.38614703445386, 11.98623388637350, 12.54465031831879, 12.89899535327330
