| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s − 3·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s − 20-s + 21-s + 9·23-s + 24-s + 25-s + 3·26-s − 27-s − 28-s − 9·29-s − 30-s + 7·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s + 1.87·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.67·29-s − 0.182·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7496011198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7496011198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.24472707084009, −15.01118290768231, −14.62747544419668, −13.39555338710184, −13.30644964866475, −12.46854813957646, −11.97463749470042, −11.59395639024034, −10.93960114121447, −10.49676673483847, −9.805974131435104, −9.452373558859883, −8.784769153629867, −8.116252647622772, −7.503511456116365, −7.024895898684522, −6.538010642456171, −5.783478470497777, −5.077429566228741, −4.619525328982019, −3.571943445399972, −3.076138875804086, −2.194386209329162, −1.245330229310216, −0.4337625663988397,
0.4337625663988397, 1.245330229310216, 2.194386209329162, 3.076138875804086, 3.571943445399972, 4.619525328982019, 5.077429566228741, 5.783478470497777, 6.538010642456171, 7.024895898684522, 7.503511456116365, 8.116252647622772, 8.784769153629867, 9.452373558859883, 9.805974131435104, 10.49676673483847, 10.93960114121447, 11.59395639024034, 11.97463749470042, 12.46854813957646, 13.30644964866475, 13.39555338710184, 14.62747544419668, 15.01118290768231, 15.24472707084009
