| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s − 13-s + 4·19-s + 4·21-s + 4·23-s − 5·25-s − 27-s − 10·29-s − 8·31-s + 2·33-s + 2·37-s + 39-s + 4·43-s − 2·47-s + 9·49-s − 2·53-s − 4·57-s − 10·59-s − 10·61-s − 4·63-s − 8·67-s − 4·69-s + 2·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.917·19-s + 0.872·21-s + 0.834·23-s − 25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s + 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.274·53-s − 0.529·57-s − 1.30·59-s − 1.28·61-s − 0.503·63-s − 0.977·67-s − 0.481·69-s + 0.237·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.19339139998143, −12.86359172704576, −12.62416065454472, −12.02552783814807, −11.49725349167678, −10.98557268506818, −10.66232763677714, −10.02489232519812, −9.515168894225638, −9.340871455491100, −8.870533755761189, −7.872443307747237, −7.484535190862502, −7.253741960887966, −6.490133840259654, −6.120423802592413, −5.444945554795519, −5.367308692311263, −4.478431090318565, −3.893579269622702, −3.287680516944157, −2.985908692532765, −2.127937010727391, −1.525791752603039, −0.5347897900215215, 0,
0.5347897900215215, 1.525791752603039, 2.127937010727391, 2.985908692532765, 3.287680516944157, 3.893579269622702, 4.478431090318565, 5.367308692311263, 5.444945554795519, 6.120423802592413, 6.490133840259654, 7.253741960887966, 7.484535190862502, 7.872443307747237, 8.870533755761189, 9.340871455491100, 9.515168894225638, 10.02489232519812, 10.66232763677714, 10.98557268506818, 11.49725349167678, 12.02552783814807, 12.62416065454472, 12.86359172704576, 13.19339139998143
