| L(s) = 1 | + (1.42 − 1.40i)2-s + (0.0543 − 3.99i)4-s − 5.95·5-s + 3.23i·7-s + (−5.54 − 5.77i)8-s + (−8.48 + 8.36i)10-s + 19.4i·11-s + 24.4·13-s + (4.54 + 4.60i)14-s + (−15.9 − 0.434i)16-s − 0.948·17-s + 4.35i·19-s + (−0.323 + 23.8i)20-s + (27.2 + 27.6i)22-s + 11.5i·23-s + ⋯ |
| L(s) = 1 | + (0.711 − 0.702i)2-s + (0.0135 − 0.999i)4-s − 1.19·5-s + 0.462i·7-s + (−0.692 − 0.721i)8-s + (−0.848 + 0.836i)10-s + 1.76i·11-s + 1.87·13-s + (0.324 + 0.329i)14-s + (−0.999 − 0.0271i)16-s − 0.0558·17-s + 0.229i·19-s + (−0.0161 + 1.19i)20-s + (1.23 + 1.25i)22-s + 0.503i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.020362204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020362204\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.42 + 1.40i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 5.95T + 25T^{2} \) |
| 7 | \( 1 - 3.23iT - 49T^{2} \) |
| 11 | \( 1 - 19.4iT - 121T^{2} \) |
| 13 | \( 1 - 24.4T + 169T^{2} \) |
| 17 | \( 1 + 0.948T + 289T^{2} \) |
| 23 | \( 1 - 11.5iT - 529T^{2} \) |
| 29 | \( 1 - 37.6T + 841T^{2} \) |
| 31 | \( 1 + 24.2iT - 961T^{2} \) |
| 37 | \( 1 - 25.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 24.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 13.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 58.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 91.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 8.80iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 148.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 192.T + 9.40e3T^{2} \) |
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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
−10.49318448540563024449528095325, −9.582751672128752714954868633467, −8.632852132246901870812748143368, −7.62929407454529582712059429338, −6.59598342443340847515589666881, −5.61477561961166474324117756507, −4.35870406013956078692355860204, −3.91370868334067736070004302341, −2.61294195598024960292970462082, −1.23160907272042267478199612930,
0.67407263190508048494997105613, 3.17936028835878126427434456482, 3.71171827291520303881473501689, 4.66395334491039645673841576540, 5.98032898865349527089143041115, 6.54417613497007012418812584294, 7.72391026490649314592332676960, 8.384678208270557476542273643586, 8.862744031032430705060084412416, 10.73569070553896881151034731117
