| L(s) = 1 | − 1.23·3-s − 2.23i·5-s − 0.521i·7-s − 7.47·9-s + 10.5i·11-s + 14.3·13-s + 2.75i·15-s − 33.4i·17-s + 23.1i·19-s + 0.643i·21-s + (−22.5 + 4.39i)23-s − 5.00·25-s + 20.3·27-s + 11.7·29-s + 17.8·31-s + ⋯ |
| L(s) = 1 | − 0.411·3-s − 0.447i·5-s − 0.0745i·7-s − 0.830·9-s + 0.962i·11-s + 1.10·13-s + 0.183i·15-s − 1.96i·17-s + 1.21i·19-s + 0.0306i·21-s + (−0.981 + 0.190i)23-s − 0.200·25-s + 0.752·27-s + 0.405·29-s + 0.575·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.209169379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209169379\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (22.5 - 4.39i)T \) |
| good | 3 | \( 1 + 1.23T + 9T^{2} \) |
| 7 | \( 1 + 0.521iT - 49T^{2} \) |
| 11 | \( 1 - 10.5iT - 121T^{2} \) |
| 13 | \( 1 - 14.3T + 169T^{2} \) |
| 17 | \( 1 + 33.4iT - 289T^{2} \) |
| 19 | \( 1 - 23.1iT - 361T^{2} \) |
| 29 | \( 1 - 11.7T + 841T^{2} \) |
| 31 | \( 1 - 17.8T + 961T^{2} \) |
| 37 | \( 1 - 57.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 53.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 14.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 85.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 31.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 39.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 30.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 49.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 96.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 56.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80.0iT - 9.40e3T^{2} \) |
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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
−8.741202589787659613755787903563, −8.271078769374676346543771273987, −7.27066459042119364019669648478, −6.45473951859591751919488998781, −5.58925144548736020999932058376, −4.93862432745969982599001252381, −3.96702746469222361128225520276, −2.89933403315843065108258651458, −1.66618874635500227322892903117, −0.41254608947918706970312038167,
0.931880792751963083851731295166, 2.35673582549591485522142334653, 3.38101811398836276276803556336, 4.16399860275209637080609961537, 5.44885193041489269123440395406, 6.16569711567819824388802069129, 6.47603249951908266263642536307, 7.83591334698592933011231931131, 8.503990353939380854343972505675, 9.000341264844348664157777383333
