| L(s) = 1 | − 3-s − 4·5-s + 7-s − 2·9-s − 2·11-s + 4·13-s + 4·15-s − 19-s − 21-s + 2·23-s + 11·25-s + 5·27-s − 9·29-s − 7·31-s + 2·33-s − 4·35-s − 6·37-s − 4·39-s + 10·41-s + 4·43-s + 8·45-s + 7·47-s + 49-s − 9·53-s + 8·55-s + 57-s − 7·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 0.377·7-s − 2/3·9-s − 0.603·11-s + 1.10·13-s + 1.03·15-s − 0.229·19-s − 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.962·27-s − 1.67·29-s − 1.25·31-s + 0.348·33-s − 0.676·35-s − 0.986·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 1.19·45-s + 1.02·47-s + 1/7·49-s − 1.23·53-s + 1.07·55-s + 0.132·57-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4544901328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4544901328\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.29737861708936, −12.93010360140045, −12.42213573470749, −11.96868365851869, −11.49545476971295, −11.04393094632187, −10.76152775831109, −10.61882578545739, −9.392272314307693, −8.998317919252711, −8.565941112509806, −8.021215678271455, −7.562704638521916, −7.261831512340494, −6.562907259431627, −5.820880322749242, −5.551373330758005, −4.877946476671375, −4.291920316050319, −3.810562509355984, −3.315977669340649, −2.732221788704939, −1.823878076621798, −0.9824366633684156, −0.2546176249464574,
0.2546176249464574, 0.9824366633684156, 1.823878076621798, 2.732221788704939, 3.315977669340649, 3.810562509355984, 4.291920316050319, 4.877946476671375, 5.551373330758005, 5.820880322749242, 6.562907259431627, 7.261831512340494, 7.562704638521916, 8.021215678271455, 8.565941112509806, 8.998317919252711, 9.392272314307693, 10.61882578545739, 10.76152775831109, 11.04393094632187, 11.49545476971295, 11.96868365851869, 12.42213573470749, 12.93010360140045, 13.29737861708936
