| L(s) = 1 | − 2·5-s − 7-s − 3·9-s − 11-s + 2·17-s + 8·23-s − 25-s − 2·29-s − 8·31-s + 2·35-s + 2·37-s − 10·41-s − 4·43-s + 6·45-s + 8·47-s + 49-s + 6·53-s + 2·55-s + 10·61-s + 3·63-s − 12·67-s + 16·71-s + 14·73-s + 77-s + 9·81-s − 4·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s − 0.301·11-s + 0.485·17-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s − 1.56·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.269·55-s + 1.28·61-s + 0.377·63-s − 1.46·67-s + 1.89·71-s + 1.63·73-s + 0.113·77-s + 81-s − 0.433·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.123836481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123836481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.98198968250827, −12.50914264069871, −12.07564204103089, −11.58188001578088, −11.20172997347882, −10.81150020677851, −10.31180027163838, −9.639591454777564, −9.254578983592839, −8.609554707033819, −8.431792653827763, −7.749247672967724, −7.283567920872325, −6.958461703017698, −6.251287809983228, −5.728191313913292, −5.111996270162468, −4.959495612406617, −3.912724948369508, −3.608184259406800, −3.176100617754049, −2.519723997436473, −1.922873172739074, −0.9407346595232828, −0.3562468144587981,
0.3562468144587981, 0.9407346595232828, 1.922873172739074, 2.519723997436473, 3.176100617754049, 3.608184259406800, 3.912724948369508, 4.959495612406617, 5.111996270162468, 5.728191313913292, 6.251287809983228, 6.958461703017698, 7.283567920872325, 7.749247672967724, 8.431792653827763, 8.609554707033819, 9.254578983592839, 9.639591454777564, 10.31180027163838, 10.81150020677851, 11.20172997347882, 11.58188001578088, 12.07564204103089, 12.50914264069871, 12.98198968250827
