| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 6·11-s − 13-s + 16-s − 6·17-s − 2·19-s + 20-s + 6·22-s − 6·23-s + 25-s − 26-s + 6·29-s − 2·31-s + 32-s − 6·34-s + 2·37-s − 2·38-s + 40-s − 6·41-s + 2·43-s + 6·44-s − 6·46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.80·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + 0.904·44-s − 0.884·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 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.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.56741957495225, −14.10332216278627, −13.66773168428078, −13.19340943235951, −12.55309703065570, −12.16316544138736, −11.58912059797294, −11.24136007244568, −10.62992827270496, −9.943464692054764, −9.560432095125895, −8.941298740480682, −8.423829261678510, −7.874856017599723, −6.864845693853483, −6.674159115037451, −6.268649721013508, −5.650596096343044, −4.790779038816361, −4.440891540168859, −3.853307946855209, −3.257554417051254, −2.356635548164774, −1.881539498070126, −1.197941505209563, 0,
1.197941505209563, 1.881539498070126, 2.356635548164774, 3.257554417051254, 3.853307946855209, 4.440891540168859, 4.790779038816361, 5.650596096343044, 6.268649721013508, 6.674159115037451, 6.864845693853483, 7.874856017599723, 8.423829261678510, 8.941298740480682, 9.560432095125895, 9.943464692054764, 10.62992827270496, 11.24136007244568, 11.58912059797294, 12.16316544138736, 12.55309703065570, 13.19340943235951, 13.66773168428078, 14.10332216278627, 14.56741957495225
