| L(s) = 1 | − 5-s − 11-s − 6·13-s − 6·17-s + 4·19-s + 25-s + 2·29-s − 8·31-s − 2·37-s + 4·41-s − 6·43-s − 8·47-s + 2·53-s + 55-s + 8·59-s + 2·61-s + 6·65-s − 8·67-s − 10·71-s − 6·73-s − 10·79-s + 16·83-s + 6·85-s + 6·89-s − 4·95-s + 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.328·37-s + 0.624·41-s − 0.914·43-s − 1.16·47-s + 0.274·53-s + 0.134·55-s + 1.04·59-s + 0.256·61-s + 0.744·65-s − 0.977·67-s − 1.18·71-s − 0.702·73-s − 1.12·79-s + 1.75·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2898816495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2898816495\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.42210520620379, −11.90698408865890, −11.66119071706556, −11.15152997757137, −10.71468858725935, −10.00512563204897, −9.977068772303249, −9.184388147562722, −8.900007809122855, −8.430364185361842, −7.692490576154230, −7.475510325217682, −7.045881568791097, −6.592611402704736, −5.956729223717048, −5.366061424200465, −4.890160387502601, −4.601507253165198, −4.010390855741554, −3.299599553887809, −2.967393637758536, −2.167041186122371, −1.956897516905058, −0.9912635821220390, −0.1495755394010171,
0.1495755394010171, 0.9912635821220390, 1.956897516905058, 2.167041186122371, 2.967393637758536, 3.299599553887809, 4.010390855741554, 4.601507253165198, 4.890160387502601, 5.366061424200465, 5.956729223717048, 6.592611402704736, 7.045881568791097, 7.475510325217682, 7.692490576154230, 8.430364185361842, 8.900007809122855, 9.184388147562722, 9.977068772303249, 10.00512563204897, 10.71468858725935, 11.15152997757137, 11.66119071706556, 11.90698408865890, 12.42210520620379
