| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s + 13-s + 16-s + 4·17-s + 18-s − 8·19-s + 22-s + 4·23-s + 24-s + 26-s + 27-s + 3·29-s + 8·31-s + 32-s + 33-s + 4·34-s + 36-s + 10·37-s − 8·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.557·29-s + 1.43·31-s + 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s + 1.64·37-s − 1.29·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.976031274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.976031274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 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
−13.82421423233167, −13.32722585050243, −12.86958732248420, −12.42999375981557, −12.04509101734078, −11.35594730004748, −10.87995823629307, −10.39446336325458, −9.971890486867121, −9.118808457567229, −8.961206765051360, −8.173093310366853, −7.780107677790906, −7.277432726616617, −6.544438003554867, −6.133175825346886, −5.751034326865948, −4.817521068838768, −4.355532645401889, −4.041757464571486, −3.230775281450016, −2.667118177262846, −2.281792899772537, −1.296821409870538, −0.7679278055440523,
0.7679278055440523, 1.296821409870538, 2.281792899772537, 2.667118177262846, 3.230775281450016, 4.041757464571486, 4.355532645401889, 4.817521068838768, 5.751034326865948, 6.133175825346886, 6.544438003554867, 7.277432726616617, 7.780107677790906, 8.173093310366853, 8.961206765051360, 9.118808457567229, 9.971890486867121, 10.39446336325458, 10.87995823629307, 11.35594730004748, 12.04509101734078, 12.42999375981557, 12.86958732248420, 13.32722585050243, 13.82421423233167
