| L(s) = 1 | − 5-s − 2·7-s + 13-s − 3·17-s + 5·23-s + 25-s − 2·29-s + 5·31-s + 2·35-s − 10·37-s + 10·43-s + 47-s − 3·49-s + 13·53-s + 5·61-s − 65-s − 8·67-s + 6·71-s + 2·73-s + 7·79-s + 4·83-s + 3·85-s − 14·89-s − 2·91-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.277·13-s − 0.727·17-s + 1.04·23-s + 1/5·25-s − 0.371·29-s + 0.898·31-s + 0.338·35-s − 1.64·37-s + 1.52·43-s + 0.145·47-s − 3/7·49-s + 1.78·53-s + 0.640·61-s − 0.124·65-s − 0.977·67-s + 0.712·71-s + 0.234·73-s + 0.787·79-s + 0.439·83-s + 0.325·85-s − 1.48·89-s − 0.209·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068322572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068322572\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.72916668063609, −12.37086167217206, −11.79271247085825, −11.37541024793459, −10.92695219599508, −10.43068612887363, −10.08513481566434, −9.404635254547573, −9.038212647874414, −8.615487721255841, −8.194090525409114, −7.489764464929410, −7.019067464774277, −6.797212091558202, −6.116284111284978, −5.695199709459438, −5.075040671737895, −4.537627783701956, −4.049035822569108, −3.439929772813567, −3.086583774376901, −2.393805167231879, −1.853514442904400, −0.9111483964413720, −0.4734805293053681,
0.4734805293053681, 0.9111483964413720, 1.853514442904400, 2.393805167231879, 3.086583774376901, 3.439929772813567, 4.049035822569108, 4.537627783701956, 5.075040671737895, 5.695199709459438, 6.116284111284978, 6.797212091558202, 7.019067464774277, 7.489764464929410, 8.194090525409114, 8.615487721255841, 9.038212647874414, 9.404635254547573, 10.08513481566434, 10.43068612887363, 10.92695219599508, 11.37541024793459, 11.79271247085825, 12.37086167217206, 12.72916668063609
