| L(s) = 1 | − 5-s − 3·9-s − 2·13-s + 2·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s + 10·37-s + 6·41-s + 4·43-s + 3·45-s − 8·47-s − 7·49-s − 6·53-s − 12·59-s + 2·61-s + 2·65-s − 8·67-s − 12·71-s + 2·73-s + 8·79-s + 9·81-s − 12·83-s − 2·85-s − 6·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.447·45-s − 1.16·47-s − 49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.42·71-s + 0.234·73-s + 0.900·79-s + 81-s − 1.31·83-s − 0.216·85-s − 0.635·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8271584545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8271584545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 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.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.93529943583421, −14.37912760824684, −13.91728064945609, −13.14711078693415, −12.74460217489228, −12.22066988479067, −11.63262699394816, −11.12296054334810, −10.85200419360842, −9.980614478580140, −9.522068403675479, −8.974965154552911, −8.323651256414687, −7.840376946929737, −7.456059843894857, −6.605314344917588, −6.018129548155382, −5.648705668810763, −4.649254596509279, −4.447232005095818, −3.473065046264370, −2.936609164335351, −2.315682084357367, −1.398319194235463, −0.3340488293757348,
0.3340488293757348, 1.398319194235463, 2.315682084357367, 2.936609164335351, 3.473065046264370, 4.447232005095818, 4.649254596509279, 5.648705668810763, 6.018129548155382, 6.605314344917588, 7.456059843894857, 7.840376946929737, 8.323651256414687, 8.974965154552911, 9.522068403675479, 9.980614478580140, 10.85200419360842, 11.12296054334810, 11.63262699394816, 12.22066988479067, 12.74460217489228, 13.14711078693415, 13.91728064945609, 14.37912760824684, 14.93529943583421
