| L(s) = 1 | − 3-s + 4·5-s + 3·7-s + 9-s + 11-s + 4·13-s − 4·15-s + 17-s + 6·19-s − 3·21-s + 2·23-s + 11·25-s − 27-s − 5·29-s − 33-s + 12·35-s − 2·37-s − 4·39-s + 9·41-s − 2·43-s + 4·45-s + 7·47-s + 2·49-s − 51-s − 9·53-s + 4·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.03·15-s + 0.242·17-s + 1.37·19-s − 0.654·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s − 0.928·29-s − 0.174·33-s + 2.02·35-s − 0.328·37-s − 0.640·39-s + 1.40·41-s − 0.304·43-s + 0.596·45-s + 1.02·47-s + 2/7·49-s − 0.140·51-s − 1.23·53-s + 0.539·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.686871215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686871215\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.75547548148379, −14.20696272390524, −13.98430284671536, −13.35890592944621, −12.95409717456722, −12.32919096942496, −11.60618275736148, −11.14282917002056, −10.79524946861915, −10.16746958148202, −9.471721506184918, −9.291391971790690, −8.523815998107011, −7.917747466337789, −7.176551258142600, −6.640761025632975, −5.945683620881325, −5.492934473226168, −5.238629257241788, −4.434810455145213, −3.659509870275757, −2.793679617646581, −1.986911716855938, −1.360151892929250, −0.9654879750576441,
0.9654879750576441, 1.360151892929250, 1.986911716855938, 2.793679617646581, 3.659509870275757, 4.434810455145213, 5.238629257241788, 5.492934473226168, 5.945683620881325, 6.640761025632975, 7.176551258142600, 7.917747466337789, 8.523815998107011, 9.291391971790690, 9.471721506184918, 10.16746958148202, 10.79524946861915, 11.14282917002056, 11.60618275736148, 12.32919096942496, 12.95409717456722, 13.35890592944621, 13.98430284671536, 14.20696272390524, 14.75547548148379
