| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s − 3·14-s − 15-s + 16-s + 2·17-s − 18-s − 5·19-s + 20-s − 3·21-s + 22-s + 4·23-s + 24-s − 4·25-s + 26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−14.79725685876617, −14.34073520695296, −13.49443684659782, −13.15049683954513, −12.55350174025299, −11.89326126630982, −11.48222127980436, −11.13147581076425, −10.46819890119450, −10.07679119041653, −9.666645621732103, −8.859188553041582, −8.455089844010181, −7.839938495334168, −7.484473004403165, −6.743510437648821, −6.187404099304898, −5.695108608649747, −5.046132104158643, −4.532150944851308, −3.921783447822493, −2.795805410784126, −2.352824108507533, −1.530736462466475, −1.009076119352324, 0,
1.009076119352324, 1.530736462466475, 2.352824108507533, 2.795805410784126, 3.921783447822493, 4.532150944851308, 5.046132104158643, 5.695108608649747, 6.187404099304898, 6.743510437648821, 7.484473004403165, 7.839938495334168, 8.455089844010181, 8.859188553041582, 9.666645621732103, 10.07679119041653, 10.46819890119450, 11.13147581076425, 11.48222127980436, 11.89326126630982, 12.55350174025299, 13.15049683954513, 13.49443684659782, 14.34073520695296, 14.79725685876617
