| L(s) = 1 | + 3·5-s − 5·11-s + 13-s − 3·17-s − 3·19-s − 3·23-s + 4·25-s + 9·29-s − 10·31-s + 37-s + 4·41-s + 13·43-s − 12·47-s + 10·53-s − 15·55-s + 6·59-s − 3·61-s + 3·65-s + 6·71-s − 9·73-s + 16·79-s − 14·83-s − 9·85-s + 4·89-s − 9·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.50·11-s + 0.277·13-s − 0.727·17-s − 0.688·19-s − 0.625·23-s + 4/5·25-s + 1.67·29-s − 1.79·31-s + 0.164·37-s + 0.624·41-s + 1.98·43-s − 1.75·47-s + 1.37·53-s − 2.02·55-s + 0.781·59-s − 0.384·61-s + 0.372·65-s + 0.712·71-s − 1.05·73-s + 1.80·79-s − 1.53·83-s − 0.976·85-s + 0.423·89-s − 0.923·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.99980562352723, −13.60135397259756, −13.04928654383330, −12.81250481194443, −12.36266463898367, −11.50872727790493, −10.97963310803513, −10.59581603447000, −10.10037512513778, −9.804967538513566, −8.924209527336308, −8.844922984004864, −8.021946488886501, −7.622640576897148, −6.879076867016969, −6.389404253179385, −5.805090337423274, −5.512226559283180, −4.834720034473965, −4.323401036280006, −3.549101506025380, −2.705457406978115, −2.306629258396301, −1.868514306383757, −0.9098571181234324, 0,
0.9098571181234324, 1.868514306383757, 2.306629258396301, 2.705457406978115, 3.549101506025380, 4.323401036280006, 4.834720034473965, 5.512226559283180, 5.805090337423274, 6.389404253179385, 6.879076867016969, 7.622640576897148, 8.021946488886501, 8.844922984004864, 8.924209527336308, 9.804967538513566, 10.10037512513778, 10.59581603447000, 10.97963310803513, 11.50872727790493, 12.36266463898367, 12.81250481194443, 13.04928654383330, 13.60135397259756, 13.99980562352723
