| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s − 21-s − 4·22-s + 8·23-s + 24-s − 25-s − 27-s + 28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927831386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927831386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−17.15864543247392, −16.78497906970211, −16.29917149144567, −15.50612205470367, −14.65492603237131, −14.49853881260461, −13.50791460179707, −13.05380978368739, −12.22516351782917, −11.47868300017097, −11.35069404796338, −10.38467665936681, −9.905065265831358, −9.157902131212734, −8.964617363597459, −7.782499075043164, −7.304133060380659, −6.509954162037661, −5.941220161892817, −5.315678531872325, −4.476956433490883, −3.471453333576736, −2.497381256428123, −1.451124959737917, −0.9267443701219515,
0.9267443701219515, 1.451124959737917, 2.497381256428123, 3.471453333576736, 4.476956433490883, 5.315678531872325, 5.941220161892817, 6.509954162037661, 7.304133060380659, 7.782499075043164, 8.964617363597459, 9.157902131212734, 9.905065265831358, 10.38467665936681, 11.35069404796338, 11.47868300017097, 12.22516351782917, 13.05380978368739, 13.50791460179707, 14.49853881260461, 14.65492603237131, 15.50612205470367, 16.29917149144567, 16.78497906970211, 17.15864543247392
