| L(s) = 1 | + 35.6·2-s + 0.238·3-s + 757.·4-s − 2.00e3·5-s + 8.48·6-s + 3.11e3·7-s + 8.74e3·8-s − 1.96e4·9-s − 7.14e4·10-s − 2.19e4·11-s + 180.·12-s − 1.70e5·13-s + 1.10e5·14-s − 477.·15-s − 7.62e4·16-s − 7.01e5·18-s + 5.78e5·19-s − 1.51e6·20-s + 740.·21-s − 7.83e5·22-s + 1.14e6·23-s + 2.08e3·24-s + 2.07e6·25-s − 6.06e6·26-s − 9.37e3·27-s + 2.35e6·28-s + 2.04e6·29-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 0.00169·3-s + 1.47·4-s − 1.43·5-s + 0.00267·6-s + 0.489·7-s + 0.754·8-s − 0.999·9-s − 2.26·10-s − 0.452·11-s + 0.00251·12-s − 1.65·13-s + 0.771·14-s − 0.00243·15-s − 0.290·16-s − 1.57·18-s + 1.01·19-s − 2.12·20-s + 0.000830·21-s − 0.713·22-s + 0.850·23-s + 0.00128·24-s + 1.06·25-s − 2.60·26-s − 0.00339·27-s + 0.724·28-s + 0.536·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.031420485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.031420485\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 35.6T + 512T^{2} \) |
| 3 | \( 1 - 0.238T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.00e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.11e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.70e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 5.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.14e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.36e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.74e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.96e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.55e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.66e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.81e6T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.42e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.40e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.49e9T + 7.60e17T^{2} \) |
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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
−10.76323557788841051642889380047, −9.207103310083448417339461450024, −7.87666208945377808498407828991, −7.36516083331671935201123116590, −6.00771223287670282870923527523, −4.89786004914804090437426293796, −4.47176501855149478445567143290, −3.12536308869171435553231160670, −2.61033748907184028418666217989, −0.57924632250111135002519711877,
0.57924632250111135002519711877, 2.61033748907184028418666217989, 3.12536308869171435553231160670, 4.47176501855149478445567143290, 4.89786004914804090437426293796, 6.00771223287670282870923527523, 7.36516083331671935201123116590, 7.87666208945377808498407828991, 9.207103310083448417339461450024, 10.76323557788841051642889380047
