| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 11-s + 6·13-s + 2·14-s + 16-s + 8·17-s − 22-s + 4·23-s + 6·26-s + 2·28-s + 10·29-s + 31-s + 32-s + 8·34-s + 2·37-s − 12·41-s − 4·43-s − 44-s + 4·46-s + 8·47-s − 3·49-s + 6·52-s − 6·53-s + 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.301·11-s + 1.66·13-s + 0.534·14-s + 1/4·16-s + 1.94·17-s − 0.213·22-s + 0.834·23-s + 1.17·26-s + 0.377·28-s + 1.85·29-s + 0.179·31-s + 0.176·32-s + 1.37·34-s + 0.328·37-s − 1.87·41-s − 0.609·43-s − 0.150·44-s + 0.589·46-s + 1.16·47-s − 3/7·49-s + 0.832·52-s − 0.824·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.467414347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.467414347\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.32870097562614, −12.96628522118770, −12.28814772358354, −11.93977584805033, −11.52289867782288, −11.01825361753146, −10.47117268588445, −10.20067114029865, −9.593305409323804, −8.760947380172808, −8.407436762097070, −8.013646254736574, −7.515730174830440, −6.793279004157410, −6.415018829287074, −5.806188511252992, −5.302860650507303, −4.932260346112980, −4.318546528148937, −3.646206568395030, −3.201262590338351, −2.762652668292417, −1.784221905351685, −1.265215179600777, −0.7885130688342966,
0.7885130688342966, 1.265215179600777, 1.784221905351685, 2.762652668292417, 3.201262590338351, 3.646206568395030, 4.318546528148937, 4.932260346112980, 5.302860650507303, 5.806188511252992, 6.415018829287074, 6.793279004157410, 7.515730174830440, 8.013646254736574, 8.407436762097070, 8.760947380172808, 9.593305409323804, 10.20067114029865, 10.47117268588445, 11.01825361753146, 11.52289867782288, 11.93977584805033, 12.28814772358354, 12.96628522118770, 13.32870097562614
