| L(s) = 1 | + 3-s + 9-s + 6·13-s + 17-s − 6·19-s + 2·23-s + 27-s + 2·29-s + 8·31-s − 4·37-s + 6·39-s − 10·41-s − 6·43-s − 8·47-s + 51-s + 12·53-s − 6·57-s − 4·59-s + 4·61-s − 10·67-s + 2·69-s + 6·73-s − 8·79-s + 81-s − 12·83-s + 2·87-s + 8·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.66·13-s + 0.242·17-s − 1.37·19-s + 0.417·23-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.657·37-s + 0.960·39-s − 1.56·41-s − 0.914·43-s − 1.16·47-s + 0.140·51-s + 1.64·53-s − 0.794·57-s − 0.520·59-s + 0.512·61-s − 1.22·67-s + 0.240·69-s + 0.702·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.214·87-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.16855002282605, −12.79897067235517, −12.09488517154754, −11.72520036003372, −11.28025295611513, −10.60407366424359, −10.35015756839532, −9.948964084603428, −9.265395702413227, −8.669554476025502, −8.402845946125658, −8.266560743184651, −7.460242992573775, −6.775593189759972, −6.576472467354664, −6.029529872635516, −5.446888810394642, −4.768156968703732, −4.347757801218458, −3.729696125534361, −3.298475465939277, −2.802157869821159, −2.031339101555833, −1.521427467498394, −0.9375375881356506, 0,
0.9375375881356506, 1.521427467498394, 2.031339101555833, 2.802157869821159, 3.298475465939277, 3.729696125534361, 4.347757801218458, 4.768156968703732, 5.446888810394642, 6.029529872635516, 6.576472467354664, 6.775593189759972, 7.460242992573775, 8.266560743184651, 8.402845946125658, 8.669554476025502, 9.265395702413227, 9.948964084603428, 10.35015756839532, 10.60407366424359, 11.28025295611513, 11.72520036003372, 12.09488517154754, 12.79897067235517, 13.16855002282605
