| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·5-s − 2·6-s + 3·8-s + 9-s + 2·10-s + 11-s − 2·12-s − 4·15-s − 16-s + 2·17-s − 18-s + 2·20-s − 22-s − 4·23-s + 6·24-s − 25-s − 4·27-s + 6·29-s + 4·30-s − 5·32-s + 2·33-s − 2·34-s − 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s − 1.03·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s − 0.213·22-s − 0.834·23-s + 1.22·24-s − 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.730·30-s − 0.883·32-s + 0.348·33-s − 0.342·34-s − 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5101163940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5101163940\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 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 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−13.81358431312704, −13.44403208012655, −13.16646286300530, −12.13063589777656, −11.95151134576394, −11.54324400660774, −10.53107119590464, −10.30995290850379, −9.837665089815965, −9.121774513408853, −8.855157751674780, −8.316033477024158, −7.973724268200014, −7.631179138455931, −6.984026348848542, −6.392493156738271, −5.534271911572395, −4.934706891380774, −4.299095052873376, −3.791565285758975, −3.352570808594692, −2.739310883996163, −1.792846858063225, −1.366749127893110, −0.2431711628280159,
0.2431711628280159, 1.366749127893110, 1.792846858063225, 2.739310883996163, 3.352570808594692, 3.791565285758975, 4.299095052873376, 4.934706891380774, 5.534271911572395, 6.392493156738271, 6.984026348848542, 7.631179138455931, 7.973724268200014, 8.316033477024158, 8.855157751674780, 9.121774513408853, 9.837665089815965, 10.30995290850379, 10.53107119590464, 11.54324400660774, 11.95151134576394, 12.13063589777656, 13.16646286300530, 13.44403208012655, 13.81358431312704
