| L(s) = 1 | − 2-s + 0.858·3-s + 4-s + 5-s − 0.858·6-s + 0.0500·7-s − 8-s − 2.26·9-s − 10-s + 11-s + 0.858·12-s + 5.55·13-s − 0.0500·14-s + 0.858·15-s + 16-s − 7.59·17-s + 2.26·18-s − 6.68·19-s + 20-s + 0.0430·21-s − 22-s − 9.07·23-s − 0.858·24-s + 25-s − 5.55·26-s − 4.51·27-s + 0.0500·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.495·3-s + 0.5·4-s + 0.447·5-s − 0.350·6-s + 0.0189·7-s − 0.353·8-s − 0.754·9-s − 0.316·10-s + 0.301·11-s + 0.247·12-s + 1.54·13-s − 0.0133·14-s + 0.221·15-s + 0.250·16-s − 1.84·17-s + 0.533·18-s − 1.53·19-s + 0.223·20-s + 0.00938·21-s − 0.213·22-s − 1.89·23-s − 0.175·24-s + 0.200·25-s − 1.09·26-s − 0.869·27-s + 0.00946·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 0.858T + 3T^{2} \) |
| 7 | \( 1 - 0.0500T + 7T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 6.68T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 - 6.11T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.778T + 53T^{2} \) |
| 59 | \( 1 - 6.37T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.77975673445486961239938570560, −6.63726577050577538008996729784, −6.18222382436653432541983856551, −5.85202313982770372278326955741, −4.30535345690172358285975425716, −4.05381444005918176036355991791, −2.67335769197217497933735983887, −2.33259498637042909934262826420, −1.30027092173428305035559780557, 0,
1.30027092173428305035559780557, 2.33259498637042909934262826420, 2.67335769197217497933735983887, 4.05381444005918176036355991791, 4.30535345690172358285975425716, 5.85202313982770372278326955741, 6.18222382436653432541983856551, 6.63726577050577538008996729784, 7.77975673445486961239938570560
