| L(s) = 1 | − 1.91i·3-s − 1.51i·5-s − 0.580·7-s − 0.682·9-s − 1.31i·11-s − 3.89i·13-s − 2.90·15-s − 1.20·17-s + i·19-s + 1.11i·21-s − 5.85·23-s + 2.70·25-s − 4.44i·27-s − 1.29i·29-s − 2.96·31-s + ⋯ |
| L(s) = 1 | − 1.10i·3-s − 0.676i·5-s − 0.219·7-s − 0.227·9-s − 0.397i·11-s − 1.07i·13-s − 0.749·15-s − 0.291·17-s + 0.229i·19-s + 0.242i·21-s − 1.22·23-s + 0.541·25-s − 0.855i·27-s − 0.239i·29-s − 0.532·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.394222 - 1.17102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.394222 - 1.17102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.91iT - 3T^{2} \) |
| 5 | \( 1 + 1.51iT - 5T^{2} \) |
| 7 | \( 1 + 0.580T + 7T^{2} \) |
| 11 | \( 1 + 1.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.89iT - 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 1.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 1.18iT - 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 - 8.38iT - 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 3.07iT - 53T^{2} \) |
| 59 | \( 1 - 0.258iT - 59T^{2} \) |
| 61 | \( 1 + 14.7iT - 61T^{2} \) |
| 67 | \( 1 + 9.54iT - 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 3.70iT - 83T^{2} \) |
| 89 | \( 1 + 8.36T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−10.29969006190725448775778879229, −9.356154294202176826375464072875, −8.242409312132469833008090386829, −7.82687782170882020688023079920, −6.67712765578991475231238496544, −5.90406129649199103529462982155, −4.81925894555596396977402893715, −3.43666974691913065286048692571, −2.01505185087395177707006491199, −0.68466440160447032079243917566,
2.15468928596932205559653942453, 3.56674694125856688858416124031, 4.30755654473878827573497985633, 5.34778481712363003774071754034, 6.60716440995334744338127145779, 7.26093437272620288710685853590, 8.646734666441536603208059665482, 9.402317770849812200498000642209, 10.17509348727234854486463352564, 10.75697754581841409050855944218
