| L(s) = 1 | + 5-s + 4·11-s + 13-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 6·29-s + 4·31-s + 6·37-s + 2·41-s − 12·43-s + 8·47-s − 7·49-s − 14·53-s + 4·55-s + 12·59-s − 2·61-s + 65-s + 8·71-s − 2·73-s − 8·79-s + 12·83-s + 2·85-s − 6·89-s + 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.277·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.312·41-s − 1.82·43-s + 1.16·47-s − 49-s − 1.92·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.124·65-s + 0.949·71-s − 0.234·73-s − 0.900·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.979721287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.979721287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.77311513150347324782512632747, −6.74326938267316721123896894436, −6.55020159931168961075059497489, −5.65940893704847595699493956150, −4.97682259964202710037865955947, −4.23599409434915576078385704417, −3.35755568876084192240443297334, −2.73375223864186048982787864682, −1.53241638521757357398128777475, −0.924917757635005337426505179673,
0.924917757635005337426505179673, 1.53241638521757357398128777475, 2.73375223864186048982787864682, 3.35755568876084192240443297334, 4.23599409434915576078385704417, 4.97682259964202710037865955947, 5.65940893704847595699493956150, 6.55020159931168961075059497489, 6.74326938267316721123896894436, 7.77311513150347324782512632747
