| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 15-s + 16-s − 4·17-s − 18-s + 2·19-s − 20-s − 4·22-s + 2·23-s − 24-s + 25-s − 26-s + 27-s + 8·29-s + 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.852·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.48·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068120051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068120051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.81231920365067, −15.12571585392420, −14.79110073708045, −14.19771171523605, −13.52142888366103, −13.08087906446457, −12.23132752071471, −11.79292705217715, −11.32270603330411, −10.64686122180531, −10.08967236000143, −9.373874508351035, −8.942033628992958, −8.495296771601188, −7.923457508004366, −7.124459292744214, −6.770308751064532, −6.166251449017905, −5.208839384253509, −4.374139459633728, −3.815759393807434, −3.085868303280991, −2.329062268140858, −1.448802603981930, −0.6918606434337525,
0.6918606434337525, 1.448802603981930, 2.329062268140858, 3.085868303280991, 3.815759393807434, 4.374139459633728, 5.208839384253509, 6.166251449017905, 6.770308751064532, 7.124459292744214, 7.923457508004366, 8.495296771601188, 8.942033628992958, 9.373874508351035, 10.08967236000143, 10.64686122180531, 11.32270603330411, 11.79292705217715, 12.23132752071471, 13.08087906446457, 13.52142888366103, 14.19771171523605, 14.79110073708045, 15.12571585392420, 15.81231920365067
