| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s − 13-s + 2·14-s − 4·16-s + 6·17-s − 2·20-s + 4·23-s − 4·25-s + 2·26-s − 2·28-s − 9·29-s + 9·31-s + 8·32-s − 12·34-s + 35-s − 4·37-s + 9·41-s − 43-s − 8·46-s + 49-s + 8·50-s − 2·52-s − 9·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s − 0.277·13-s + 0.534·14-s − 16-s + 1.45·17-s − 0.447·20-s + 0.834·23-s − 4/5·25-s + 0.392·26-s − 0.377·28-s − 1.67·29-s + 1.61·31-s + 1.41·32-s − 2.05·34-s + 0.169·35-s − 0.657·37-s + 1.40·41-s − 0.152·43-s − 1.17·46-s + 1/7·49-s + 1.13·50-s − 0.277·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−12.77553609557539, −12.29160928619380, −12.01048894526007, −11.35212359597415, −10.97318302347049, −10.62904042654571, −9.983031554398617, −9.518618290718178, −9.505811971965765, −8.819915130212179, −8.235974194853664, −7.811900475573389, −7.635420903564132, −7.013627563785837, −6.620723250588306, −5.894153119585929, −5.510882025733275, −4.805592100050931, −4.266897414436892, −3.700361972385282, −3.053451687484094, −2.623037681147846, −1.771344982549337, −1.307260303182898, −0.6309828939466900, 0,
0.6309828939466900, 1.307260303182898, 1.771344982549337, 2.623037681147846, 3.053451687484094, 3.700361972385282, 4.266897414436892, 4.805592100050931, 5.510882025733275, 5.894153119585929, 6.620723250588306, 7.013627563785837, 7.635420903564132, 7.811900475573389, 8.235974194853664, 8.819915130212179, 9.505811971965765, 9.518618290718178, 9.983031554398617, 10.62904042654571, 10.97318302347049, 11.35212359597415, 12.01048894526007, 12.29160928619380, 12.77553609557539
