| L(s) = 1 | + 3-s − 2·5-s + 9-s − 3·11-s − 13-s − 2·15-s + 5·17-s + 7·19-s − 6·23-s − 25-s + 27-s + 5·29-s + 8·31-s − 3·33-s + 10·37-s − 39-s + 4·41-s + 12·43-s − 2·45-s + 7·47-s + 5·51-s − 9·53-s + 6·55-s + 7·57-s + 3·59-s − 11·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.516·15-s + 1.21·17-s + 1.60·19-s − 1.25·23-s − 1/5·25-s + 0.192·27-s + 0.928·29-s + 1.43·31-s − 0.522·33-s + 1.64·37-s − 0.160·39-s + 0.624·41-s + 1.82·43-s − 0.298·45-s + 1.02·47-s + 0.700·51-s − 1.23·53-s + 0.809·55-s + 0.927·57-s + 0.390·59-s − 1.40·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.20232691683673, −12.38083742510139, −12.25446217651576, −11.86292710403410, −11.32855652771474, −10.76602794894337, −10.16702673870840, −9.926417354420402, −9.415234181779028, −8.918567025237908, −8.181962086007924, −7.797448121663562, −7.641855655624423, −7.351361418503872, −6.369722344147795, −5.921829490993057, −5.496790060295541, −4.667555370524808, −4.433961090545492, −3.797087135557382, −3.209717387657612, −2.719829763516585, −2.400772538814545, −1.270107572393208, −0.9036766829636705, 0,
0.9036766829636705, 1.270107572393208, 2.400772538814545, 2.719829763516585, 3.209717387657612, 3.797087135557382, 4.433961090545492, 4.667555370524808, 5.496790060295541, 5.921829490993057, 6.369722344147795, 7.351361418503872, 7.641855655624423, 7.797448121663562, 8.181962086007924, 8.918567025237908, 9.415234181779028, 9.926417354420402, 10.16702673870840, 10.76602794894337, 11.32855652771474, 11.86292710403410, 12.25446217651576, 12.38083742510139, 13.20232691683673
