| L(s) = 1 | − 2·4-s + 5-s + 2·7-s + 13-s + 4·16-s − 3·17-s + 2·19-s − 2·20-s + 23-s + 25-s − 4·28-s + 4·29-s + 3·31-s + 2·35-s − 8·37-s + 5·41-s + 43-s − 3·49-s − 2·52-s + 5·53-s − 12·59-s + 4·61-s − 8·64-s + 65-s − 3·67-s + 6·68-s − 6·71-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.755·7-s + 0.277·13-s + 16-s − 0.727·17-s + 0.458·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s − 0.755·28-s + 0.742·29-s + 0.538·31-s + 0.338·35-s − 1.31·37-s + 0.780·41-s + 0.152·43-s − 3/7·49-s − 0.277·52-s + 0.686·53-s − 1.56·59-s + 0.512·61-s − 64-s + 0.124·65-s − 0.366·67-s + 0.727·68-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.20371966591168, −12.78153762608746, −12.18453261746942, −11.89888749108083, −11.20291305090345, −10.77223547680512, −10.37945138988649, −9.831002186286389, −9.357540977431775, −8.917791264098852, −8.539523512359338, −8.034713165427282, −7.648341702827530, −6.893721425067567, −6.511683624147774, −5.817423599541812, −5.339805969465335, −4.993648314409139, −4.285490521040319, −4.161559206294653, −3.216350664270939, −2.834864350744665, −1.985032893424278, −1.415227007169979, −0.8466874714664820, 0,
0.8466874714664820, 1.415227007169979, 1.985032893424278, 2.834864350744665, 3.216350664270939, 4.161559206294653, 4.285490521040319, 4.993648314409139, 5.339805969465335, 5.817423599541812, 6.511683624147774, 6.893721425067567, 7.648341702827530, 8.034713165427282, 8.539523512359338, 8.917791264098852, 9.357540977431775, 9.831002186286389, 10.37945138988649, 10.77223547680512, 11.20291305090345, 11.89888749108083, 12.18453261746942, 12.78153762608746, 13.20371966591168
