| L(s) = 1 | − 2·3-s + 5-s + 9-s − 11-s − 4·13-s − 2·15-s + 8·17-s − 8·19-s − 4·23-s + 25-s + 4·27-s + 2·29-s + 2·31-s + 2·33-s + 6·37-s + 8·39-s + 12·41-s + 4·43-s + 45-s − 6·47-s − 16·51-s + 6·53-s − 55-s + 16·57-s − 6·59-s − 12·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.516·15-s + 1.94·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.986·37-s + 1.28·39-s + 1.87·41-s + 0.609·43-s + 0.149·45-s − 0.875·47-s − 2.24·51-s + 0.824·53-s − 0.134·55-s + 2.11·57-s − 0.781·59-s − 1.53·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.27456545320179, −12.81911219468533, −12.38826183549805, −12.01673867429349, −11.75996729170307, −10.90234046679427, −10.64496189599835, −10.23344769315271, −9.794529735448412, −9.286340251400018, −8.674242452936610, −8.010515641536514, −7.625750761410495, −7.183179581932183, −6.308617532083493, −6.015905109148530, −5.839478065530149, −5.054598066879250, −4.624886695398011, −4.232801961467969, −3.326206358203046, −2.680212847666249, −2.228738937331752, −1.385175381672251, −0.6967438679656641, 0,
0.6967438679656641, 1.385175381672251, 2.228738937331752, 2.680212847666249, 3.326206358203046, 4.232801961467969, 4.624886695398011, 5.054598066879250, 5.839478065530149, 6.015905109148530, 6.308617532083493, 7.183179581932183, 7.625750761410495, 8.010515641536514, 8.674242452936610, 9.286340251400018, 9.794529735448412, 10.23344769315271, 10.64496189599835, 10.90234046679427, 11.75996729170307, 12.01673867429349, 12.38826183549805, 12.81911219468533, 13.27456545320179
