| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 2·12-s + 13-s + 4·14-s + 16-s + 6·17-s − 18-s + 8·19-s + 8·21-s + 7·23-s + 2·24-s − 26-s + 4·27-s − 4·28-s − 6·29-s − 7·31-s − 32-s − 6·34-s + 36-s − 4·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.83·19-s + 1.74·21-s + 1.45·23-s + 0.408·24-s − 0.196·26-s + 0.769·27-s − 0.755·28-s − 1.11·29-s − 1.25·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.657·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9035259937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9035259937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 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
−14.16626457222905, −13.26409755472136, −12.79562520814272, −12.56573018942586, −11.88012971692614, −11.46873787143324, −11.03470786834500, −10.53165601392577, −9.828533560021829, −9.698918154586223, −9.086943126278198, −8.629928023023377, −7.630107142572390, −7.315960927323655, −6.937989899478188, −6.154421446725597, −5.791601465428662, −5.405262968803983, −4.815486728540428, −3.662424752942807, −3.281816220476683, −2.885306765551903, −1.731901735655829, −0.9225254170988050, −0.4950309338499874,
0.4950309338499874, 0.9225254170988050, 1.731901735655829, 2.885306765551903, 3.281816220476683, 3.662424752942807, 4.815486728540428, 5.405262968803983, 5.791601465428662, 6.154421446725597, 6.937989899478188, 7.315960927323655, 7.630107142572390, 8.629928023023377, 9.086943126278198, 9.698918154586223, 9.828533560021829, 10.53165601392577, 11.03470786834500, 11.46873787143324, 11.88012971692614, 12.56573018942586, 12.79562520814272, 13.26409755472136, 14.16626457222905
