| L(s) = 1 | + 3-s + 2·7-s + 9-s − 2·11-s − 13-s + 4·19-s + 2·21-s − 8·23-s − 5·25-s + 27-s − 2·29-s − 6·31-s − 2·33-s − 8·37-s − 39-s − 4·41-s + 4·43-s − 3·49-s − 14·53-s + 4·57-s − 8·59-s + 10·61-s + 2·63-s + 12·67-s − 8·69-s − 6·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.917·19-s + 0.436·21-s − 1.66·23-s − 25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.348·33-s − 1.31·37-s − 0.160·39-s − 0.624·41-s + 0.609·43-s − 3/7·49-s − 1.92·53-s + 0.529·57-s − 1.04·59-s + 1.28·61-s + 0.251·63-s + 1.46·67-s − 0.963·69-s − 0.712·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543125289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543125289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.20491793174913, −12.74945927435413, −12.08163543458823, −11.93136052963207, −11.19081056488921, −10.87714024856523, −10.23462968480822, −9.805572970123910, −9.399486109318276, −8.902624605478367, −8.108849795693152, −7.965708201688175, −7.622666575318246, −6.956319113154124, −6.420063455959366, −5.662978006346470, −5.307867137608808, −4.835423137425694, −4.095320008043894, −3.672689160239103, −3.131193226160880, −2.355585329566976, −1.857522334715775, −1.461757739692554, −0.3187096510417091,
0.3187096510417091, 1.461757739692554, 1.857522334715775, 2.355585329566976, 3.131193226160880, 3.672689160239103, 4.095320008043894, 4.835423137425694, 5.307867137608808, 5.662978006346470, 6.420063455959366, 6.956319113154124, 7.622666575318246, 7.965708201688175, 8.108849795693152, 8.902624605478367, 9.399486109318276, 9.805572970123910, 10.23462968480822, 10.87714024856523, 11.19081056488921, 11.93136052963207, 12.08163543458823, 12.74945927435413, 13.20491793174913
