| L(s) = 1 | − 2.20·2-s − 3.23·3-s + 2.85·4-s − 3.16·5-s + 7.13·6-s + 0.235·7-s − 1.88·8-s + 7.49·9-s + 6.97·10-s − 3.79·11-s − 9.24·12-s − 0.519·14-s + 10.2·15-s − 1.55·16-s + 4.32·17-s − 16.5·18-s + 1.70·19-s − 9.04·20-s − 0.763·21-s + 8.35·22-s + 23-s + 6.10·24-s + 5.02·25-s − 14.5·27-s + 0.673·28-s + 1.21·29-s − 22.6·30-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 1.87·3-s + 1.42·4-s − 1.41·5-s + 2.91·6-s + 0.0891·7-s − 0.666·8-s + 2.49·9-s + 2.20·10-s − 1.14·11-s − 2.67·12-s − 0.138·14-s + 2.64·15-s − 0.389·16-s + 1.04·17-s − 3.89·18-s + 0.391·19-s − 2.02·20-s − 0.166·21-s + 1.78·22-s + 0.208·23-s + 1.24·24-s + 1.00·25-s − 2.80·27-s + 0.127·28-s + 0.224·29-s − 4.12·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2715144278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2715144278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 0.235T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 29 | \( 1 - 1.21T + 29T^{2} \) |
| 31 | \( 1 - 9.90T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.579T + 83T^{2} \) |
| 89 | \( 1 + 2.75T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−8.121341820886830350618223516687, −7.81986740810124191164517785144, −7.26806344628686235554825876697, −6.47945167160882568261106956062, −5.63091831691367838998227153536, −4.79391214967684296210437756696, −4.11565958591798461020618132957, −2.70779083180638192973882295032, −1.08744131296642473215478477949, −0.53597146091981117016397617633,
0.53597146091981117016397617633, 1.08744131296642473215478477949, 2.70779083180638192973882295032, 4.11565958591798461020618132957, 4.79391214967684296210437756696, 5.63091831691367838998227153536, 6.47945167160882568261106956062, 7.26806344628686235554825876697, 7.81986740810124191164517785144, 8.121341820886830350618223516687
