| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s − 2·7-s + 8-s + 9-s + 4·10-s + 11-s + 12-s − 2·14-s + 4·15-s + 16-s − 17-s + 18-s + 4·20-s − 2·21-s + 22-s − 6·23-s + 24-s + 11·25-s + 27-s − 2·28-s − 2·29-s + 4·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.894·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.730·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.827933235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.827933235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−19.48484011569538, −18.46192536321198, −17.95716393510138, −17.05010588842074, −16.50555223403942, −15.68208021244174, −14.81685355416984, −14.19732053129507, −13.54866702540893, −13.20150895181096, −12.44455790920466, −11.50293441178465, −10.25516847678640, −9.957781387417194, −9.184187584794878, −8.288845053216673, −7.000373650045399, −6.327651793952685, −5.711403571184931, −4.664355373423064, −3.480479945649647, −2.519433271965645, −1.653973356462181,
1.653973356462181, 2.519433271965645, 3.480479945649647, 4.664355373423064, 5.711403571184931, 6.327651793952685, 7.000373650045399, 8.288845053216673, 9.184187584794878, 9.957781387417194, 10.25516847678640, 11.50293441178465, 12.44455790920466, 13.20150895181096, 13.54866702540893, 14.19732053129507, 14.81685355416984, 15.68208021244174, 16.50555223403942, 17.05010588842074, 17.95716393510138, 18.46192536321198, 19.48484011569538
