| L(s) = 1 | − 5-s − 11-s − 2·13-s − 6·17-s − 4·19-s + 25-s + 6·29-s − 4·31-s + 2·37-s − 6·41-s + 4·43-s + 6·53-s + 55-s − 12·59-s − 14·61-s + 2·65-s + 16·67-s − 2·73-s + 4·79-s + 6·85-s − 6·89-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s + 0.134·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s + 1.95·67-s − 0.234·73-s + 0.450·79-s + 0.650·85-s − 0.635·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3602351076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3602351076\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.54686629576508, −12.08546778930320, −11.46018342434510, −11.14615517208039, −10.64519255009542, −10.34742072586884, −9.777459698349358, −9.108969706657086, −8.940932803739249, −8.319527815172540, −7.959215853972619, −7.422212522089257, −6.923336655007380, −6.495278813463046, −6.121571721524139, −5.374012775832533, −4.919094414377948, −4.402460059441723, −4.123990550053124, −3.402847473526634, −2.815677652410337, −2.318394530154636, −1.837805284136731, −1.023508187993305, −0.1645029300605285,
0.1645029300605285, 1.023508187993305, 1.837805284136731, 2.318394530154636, 2.815677652410337, 3.402847473526634, 4.123990550053124, 4.402460059441723, 4.919094414377948, 5.374012775832533, 6.121571721524139, 6.495278813463046, 6.923336655007380, 7.422212522089257, 7.959215853972619, 8.319527815172540, 8.940932803739249, 9.108969706657086, 9.777459698349358, 10.34742072586884, 10.64519255009542, 11.14615517208039, 11.46018342434510, 12.08546778930320, 12.54686629576508
