| L(s) = 1 | + 4·13-s − 17-s + 2·19-s + 9·23-s − 5·25-s + 9·29-s + 5·31-s + 2·37-s + 43-s + 9·47-s − 9·59-s + 61-s − 5·67-s − 9·71-s + 7·73-s + 10·79-s − 9·89-s + 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 0.242·17-s + 0.458·19-s + 1.87·23-s − 25-s + 1.67·29-s + 0.898·31-s + 0.328·37-s + 0.152·43-s + 1.31·47-s − 1.17·59-s + 0.128·61-s − 0.610·67-s − 1.06·71-s + 0.819·73-s + 1.12·79-s − 0.953·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 359856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 359856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.808765808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.808765808\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.33790517057515, −12.16582041105678, −11.62667080566845, −11.04038680958227, −10.85534209433622, −10.30011165506848, −9.828781900030745, −9.246221586683815, −8.914170966423094, −8.448793525594840, −7.981839529027217, −7.484832431838689, −6.941785458037014, −6.482255256800641, −6.057771711899950, −5.564127757701324, −4.961296716017968, −4.494189661509677, −4.038166623784782, −3.347914618323136, −2.923058715109286, −2.446054507517353, −1.589175452413315, −1.057896682054789, −0.5824374539182112,
0.5824374539182112, 1.057896682054789, 1.589175452413315, 2.446054507517353, 2.923058715109286, 3.347914618323136, 4.038166623784782, 4.494189661509677, 4.961296716017968, 5.564127757701324, 6.057771711899950, 6.482255256800641, 6.941785458037014, 7.484832431838689, 7.981839529027217, 8.448793525594840, 8.914170966423094, 9.246221586683815, 9.828781900030745, 10.30011165506848, 10.85534209433622, 11.04038680958227, 11.62667080566845, 12.16582041105678, 12.33790517057515
