| L(s) = 1 | − 5-s + 4·7-s − 11-s − 13-s + 4·17-s − 6·19-s − 4·23-s + 25-s − 2·29-s − 4·31-s − 4·35-s + 4·37-s + 10·41-s − 4·43-s − 8·47-s + 9·49-s − 2·53-s + 55-s − 4·59-s + 14·61-s + 65-s + 2·67-s − 12·71-s + 2·73-s − 4·77-s − 8·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.970·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.657·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.134·55-s − 0.520·59-s + 1.79·61-s + 0.124·65-s + 0.244·67-s − 1.42·71-s + 0.234·73-s − 0.455·77-s − 0.900·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 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.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.24155670666865, −13.40162837521492, −12.98183485639105, −12.45959595818132, −11.97312373484521, −11.48760369939529, −11.09753223688565, −10.60758383042046, −10.17426326216356, −9.489269480741728, −8.983541377428245, −8.271685969600750, −7.940148119100594, −7.760095769181982, −6.993111582306926, −6.435241119903814, −5.601044534653789, −5.417269448766343, −4.530490179424333, −4.339321481250243, −3.661009038597264, −2.907995397416966, −2.132418033532113, −1.733977786576069, −0.8942928872295801, 0,
0.8942928872295801, 1.733977786576069, 2.132418033532113, 2.907995397416966, 3.661009038597264, 4.339321481250243, 4.530490179424333, 5.417269448766343, 5.601044534653789, 6.435241119903814, 6.993111582306926, 7.760095769181982, 7.940148119100594, 8.271685969600750, 8.983541377428245, 9.489269480741728, 10.17426326216356, 10.60758383042046, 11.09753223688565, 11.48760369939529, 11.97312373484521, 12.45959595818132, 12.98183485639105, 13.40162837521492, 14.24155670666865
