| L(s) = 1 | + 5-s − 11-s + 2·13-s + 2·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s − 2·37-s − 6·41-s + 4·43-s + 8·47-s + 10·53-s − 55-s − 4·59-s + 2·61-s + 2·65-s − 12·67-s − 8·71-s + 14·73-s + 8·79-s + 12·83-s + 2·85-s − 6·89-s + 4·95-s + 14·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1.37·53-s − 0.134·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.63·73-s + 0.900·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.039915864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039915864\) |
| \(L(\frac{3}{2})\) |
|
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.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.23444722834823, −12.53693548130626, −12.17578869977361, −11.71125370505957, −11.29449721533574, −10.56691172990651, −10.25311557825354, −9.961510639465076, −9.217328198850361, −8.885014941186100, −8.379231015061609, −7.787353333691583, −7.382962852042773, −6.838219927017832, −6.162594261533585, −5.908723903964543, −5.267049251974072, −4.804839910806948, −4.260971533134908, −3.441744563702931, −3.177950891016350, −2.411006956581000, −1.899085854403994, −1.023082922036512, −0.6765333479311073,
0.6765333479311073, 1.023082922036512, 1.899085854403994, 2.411006956581000, 3.177950891016350, 3.441744563702931, 4.260971533134908, 4.804839910806948, 5.267049251974072, 5.908723903964543, 6.162594261533585, 6.838219927017832, 7.382962852042773, 7.787353333691583, 8.379231015061609, 8.885014941186100, 9.217328198850361, 9.961510639465076, 10.25311557825354, 10.56691172990651, 11.29449721533574, 11.71125370505957, 12.17578869977361, 12.53693548130626, 13.23444722834823
