| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 4·13-s + 21-s + 4·23-s − 27-s + 2·29-s − 4·31-s + 33-s − 8·37-s + 4·39-s − 2·41-s + 8·43-s + 12·47-s + 49-s + 4·53-s − 12·59-s + 2·61-s − 63-s − 4·67-s − 4·69-s − 8·71-s + 12·73-s + 77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s − 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.549·53-s − 1.56·59-s + 0.256·61-s − 0.125·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.40·73-s + 0.113·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.69027679683811, −15.32073953628611, −14.74500337581255, −14.04247610694573, −13.63748099288277, −12.86003386739192, −12.43028484860940, −12.10796881032334, −11.37084209638074, −10.75663402616314, −10.33763750535456, −9.800186247943418, −9.032852908992300, −8.760205202415578, −7.562214316823654, −7.442308365454713, −6.732984508238772, −6.057482461677654, −5.411492135374102, −4.918237060367551, −4.248531273026684, −3.433543452243221, −2.693616616576539, −1.968869667773541, −0.8957042270534924, 0,
0.8957042270534924, 1.968869667773541, 2.693616616576539, 3.433543452243221, 4.248531273026684, 4.918237060367551, 5.411492135374102, 6.057482461677654, 6.732984508238772, 7.442308365454713, 7.562214316823654, 8.760205202415578, 9.032852908992300, 9.800186247943418, 10.33763750535456, 10.75663402616314, 11.37084209638074, 12.10796881032334, 12.43028484860940, 12.86003386739192, 13.63748099288277, 14.04247610694573, 14.74500337581255, 15.32073953628611, 15.69027679683811
