| L(s) = 1 | + 2·3-s − 3·5-s + 9-s − 6·15-s + 17-s − 2·19-s + 6·23-s + 4·25-s − 4·27-s − 3·29-s − 2·31-s + 3·37-s + 7·41-s − 8·43-s − 3·45-s − 2·47-s + 2·51-s + 5·53-s − 4·57-s + 3·61-s + 2·67-s + 12·69-s + 10·71-s + 11·73-s + 8·75-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 1/3·9-s − 1.54·15-s + 0.242·17-s − 0.458·19-s + 1.25·23-s + 4/5·25-s − 0.769·27-s − 0.557·29-s − 0.359·31-s + 0.493·37-s + 1.09·41-s − 1.21·43-s − 0.447·45-s − 0.291·47-s + 0.280·51-s + 0.686·53-s − 0.529·57-s + 0.384·61-s + 0.244·67-s + 1.44·69-s + 1.18·71-s + 1.28·73-s + 0.923·75-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351116158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351116158\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.88001341102966, −12.37669406035662, −11.84253196438923, −11.46228013035961, −10.88909280725652, −10.73809525078671, −9.793799816331973, −9.514526632403467, −8.956769573973015, −8.569562653248315, −8.080611293197550, −7.808100743919489, −7.314964428122523, −6.846095381324295, −6.318123869108338, −5.502807461303164, −5.103463561995528, −4.359019235625537, −4.006067614125148, −3.391701513877644, −3.200178219744525, −2.430778199652233, −1.987727156917172, −1.078332608026750, −0.4180946717317457,
0.4180946717317457, 1.078332608026750, 1.987727156917172, 2.430778199652233, 3.200178219744525, 3.391701513877644, 4.006067614125148, 4.359019235625537, 5.103463561995528, 5.502807461303164, 6.318123869108338, 6.846095381324295, 7.314964428122523, 7.808100743919489, 8.080611293197550, 8.569562653248315, 8.956769573973015, 9.514526632403467, 9.793799816331973, 10.73809525078671, 10.88909280725652, 11.46228013035961, 11.84253196438923, 12.37669406035662, 12.88001341102966
