| L(s) = 1 | + 2-s − 4-s − 3·8-s − 11-s + 4·13-s − 16-s − 4·17-s − 22-s − 4·23-s + 4·26-s + 6·29-s − 10·31-s + 5·32-s − 4·34-s + 6·37-s + 4·41-s − 12·43-s + 44-s − 4·46-s + 10·47-s − 4·52-s − 6·53-s + 6·58-s + 2·59-s − 10·62-s + 7·64-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.301·11-s + 1.10·13-s − 1/4·16-s − 0.970·17-s − 0.213·22-s − 0.834·23-s + 0.784·26-s + 1.11·29-s − 1.79·31-s + 0.883·32-s − 0.685·34-s + 0.986·37-s + 0.624·41-s − 1.82·43-s + 0.150·44-s − 0.589·46-s + 1.45·47-s − 0.554·52-s − 0.824·53-s + 0.787·58-s + 0.260·59-s − 1.27·62-s + 7/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.76736859505414, −13.33384447267675, −12.94187013152244, −12.48559990267212, −12.00669867685357, −11.36781944270972, −11.03034361747115, −10.46411343663098, −9.886578072053579, −9.348757244993429, −8.862365992733141, −8.456905023231455, −7.989911865793499, −7.313238704936703, −6.637886655643196, −6.154596299611079, −5.765871269974483, −5.168665293861473, −4.581909454051484, −4.122406940941044, −3.628734171267754, −3.061702135388496, −2.370826349460062, −1.678235857890743, −0.7760465689535254, 0,
0.7760465689535254, 1.678235857890743, 2.370826349460062, 3.061702135388496, 3.628734171267754, 4.122406940941044, 4.581909454051484, 5.168665293861473, 5.765871269974483, 6.154596299611079, 6.637886655643196, 7.313238704936703, 7.989911865793499, 8.456905023231455, 8.862365992733141, 9.348757244993429, 9.886578072053579, 10.46411343663098, 11.03034361747115, 11.36781944270972, 12.00669867685357, 12.48559990267212, 12.94187013152244, 13.33384447267675, 13.76736859505414
