| L(s) = 1 | − 5-s + 2·13-s + 2·17-s + 8·23-s + 25-s + 2·29-s + 8·31-s − 4·37-s − 8·41-s − 4·43-s − 8·47-s + 12·53-s + 4·59-s + 6·61-s − 2·65-s + 4·67-s + 14·71-s + 2·73-s + 2·79-s + 6·83-s − 2·85-s + 12·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s + 0.485·17-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.657·37-s − 1.24·41-s − 0.609·43-s − 1.16·47-s + 1.64·53-s + 0.520·59-s + 0.768·61-s − 0.248·65-s + 0.488·67-s + 1.66·71-s + 0.234·73-s + 0.225·79-s + 0.658·83-s − 0.216·85-s + 1.27·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.119316135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.119316135\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.80379440684918, −12.20737869523297, −11.75641634024310, −11.47306756061115, −10.95641754694866, −10.38058078682817, −10.09454690541022, −9.556769406374124, −8.893494571390281, −8.567930068306519, −8.158917559976711, −7.678372933238392, −7.003444326437591, −6.652435902773290, −6.330676789271302, −5.446726525058625, −5.074706815860644, −4.749737208653662, −3.888566862450361, −3.567562176911759, −3.023816358614016, −2.460257227589361, −1.700080762172580, −0.9956296497751747, −0.5671382292797426,
0.5671382292797426, 0.9956296497751747, 1.700080762172580, 2.460257227589361, 3.023816358614016, 3.567562176911759, 3.888566862450361, 4.749737208653662, 5.074706815860644, 5.446726525058625, 6.330676789271302, 6.652435902773290, 7.003444326437591, 7.678372933238392, 8.158917559976711, 8.567930068306519, 8.893494571390281, 9.556769406374124, 10.09454690541022, 10.38058078682817, 10.95641754694866, 11.47306756061115, 11.75641634024310, 12.20737869523297, 12.80379440684918
