| L(s) = 1 | − 3·5-s + 2·7-s + 3·11-s + 4·13-s + 7·19-s + 3·23-s + 4·25-s + 6·29-s + 8·31-s − 6·35-s + 11·37-s + 4·43-s − 12·47-s − 3·49-s − 3·53-s − 9·55-s + 9·59-s − 61-s − 12·65-s + 13·67-s + 6·71-s + 10·73-s + 6·77-s − 8·79-s + 3·89-s + 8·91-s − 21·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s + 0.904·11-s + 1.10·13-s + 1.60·19-s + 0.625·23-s + 4/5·25-s + 1.11·29-s + 1.43·31-s − 1.01·35-s + 1.80·37-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.412·53-s − 1.21·55-s + 1.17·59-s − 0.128·61-s − 1.48·65-s + 1.58·67-s + 0.712·71-s + 1.17·73-s + 0.683·77-s − 0.900·79-s + 0.317·89-s + 0.838·91-s − 2.15·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.900600606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.900600606\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 83 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.79361056937112, −12.27569941517440, −11.73237850132013, −11.47516342608958, −11.23883004664552, −10.82163713086095, −9.886540057265651, −9.737646895324131, −9.042512626166982, −8.484226385082263, −8.082486126929219, −7.904878359793518, −7.266632623280799, −6.605342416227464, −6.420848296653014, −5.645759187517050, −4.963965289422579, −4.656730609096063, −4.032266918487558, −3.656117219901374, −3.089244082620823, −2.566607781016830, −1.537595389206382, −0.9949637890949773, −0.7014044597262417,
0.7014044597262417, 0.9949637890949773, 1.537595389206382, 2.566607781016830, 3.089244082620823, 3.656117219901374, 4.032266918487558, 4.656730609096063, 4.963965289422579, 5.645759187517050, 6.420848296653014, 6.605342416227464, 7.266632623280799, 7.904878359793518, 8.082486126929219, 8.484226385082263, 9.042512626166982, 9.737646895324131, 9.886540057265651, 10.82163713086095, 11.23883004664552, 11.47516342608958, 11.73237850132013, 12.27569941517440, 12.79361056937112
