| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·11-s + 2·13-s + 2·15-s − 2·17-s − 19-s − 6·23-s − 25-s − 27-s + 8·31-s + 2·33-s − 6·37-s − 2·39-s + 8·43-s − 2·45-s − 6·47-s + 2·51-s + 4·55-s + 57-s + 10·61-s − 4·65-s + 4·67-s + 6·69-s − 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.229·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.348·33-s − 0.986·37-s − 0.320·39-s + 1.21·43-s − 0.298·45-s − 0.875·47-s + 0.280·51-s + 0.539·55-s + 0.132·57-s + 1.28·61-s − 0.496·65-s + 0.488·67-s + 0.722·69-s − 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.29758705362763, −12.98204207590903, −12.25409841921266, −11.98275665862107, −11.60306302034866, −11.07389636669645, −10.61874996757414, −10.23586850830434, −9.706020980475505, −9.123210907464760, −8.389191676774325, −8.192622106567543, −7.693936498893061, −7.122760970372709, −6.551586946465333, −6.173659350401754, −5.537485733904894, −5.095179596216700, −4.294378068847520, −4.150654141055426, −3.497226256050126, −2.793620923486589, −2.162677908720092, −1.449390008728529, −0.6101873945346485, 0,
0.6101873945346485, 1.449390008728529, 2.162677908720092, 2.793620923486589, 3.497226256050126, 4.150654141055426, 4.294378068847520, 5.095179596216700, 5.537485733904894, 6.173659350401754, 6.551586946465333, 7.122760970372709, 7.693936498893061, 8.192622106567543, 8.389191676774325, 9.123210907464760, 9.706020980475505, 10.23586850830434, 10.61874996757414, 11.07389636669645, 11.60306302034866, 11.98275665862107, 12.25409841921266, 12.98204207590903, 13.29758705362763
