| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s + 4·13-s + 16-s + 20-s + 4·22-s + 4·23-s + 25-s + 4·26-s + 6·29-s + 8·31-s + 32-s + 2·37-s + 40-s − 10·41-s − 8·43-s + 4·44-s + 4·46-s + 6·47-s − 7·49-s + 50-s + 4·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.10·13-s + 1/4·16-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.328·37-s + 0.158·40-s − 1.56·41-s − 1.21·43-s + 0.603·44-s + 0.589·46-s + 0.875·47-s − 49-s + 0.141·50-s + 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.492257693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.492257693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.920744652214994132989169808967, −6.92503454753842892323309327695, −6.35869810777261212776647654333, −6.05456129068156884171413937704, −4.92454276512485882449405911672, −4.49725664476061102252678083122, −3.47759773004767530387464492411, −2.99058818549007676826487674667, −1.75734105359211111866437132781, −1.07041918042935000282290691609,
1.07041918042935000282290691609, 1.75734105359211111866437132781, 2.99058818549007676826487674667, 3.47759773004767530387464492411, 4.49725664476061102252678083122, 4.92454276512485882449405911672, 6.05456129068156884171413937704, 6.35869810777261212776647654333, 6.92503454753842892323309327695, 7.920744652214994132989169808967
