| L(s) = 1 | + 2-s − 3-s + 4-s + 0.491·5-s − 6-s − 7-s + 8-s + 9-s + 0.491·10-s − 3.78·11-s − 12-s − 4.72·13-s − 14-s − 0.491·15-s + 16-s + 3.78·17-s + 18-s + 1.59·19-s + 0.491·20-s + 21-s − 3.78·22-s + 2.70·23-s − 24-s − 4.75·25-s − 4.72·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.219·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.155·10-s − 1.14·11-s − 0.288·12-s − 1.31·13-s − 0.267·14-s − 0.126·15-s + 0.250·16-s + 0.917·17-s + 0.235·18-s + 0.366·19-s + 0.109·20-s + 0.218·21-s − 0.806·22-s + 0.564·23-s − 0.204·24-s − 0.951·25-s − 0.926·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 0.491T + 5T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 - 5.17T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 0.0475T + 37T^{2} \) |
| 41 | \( 1 - 0.784T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 3.59T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.53154645365425419861079550746, −6.69934542100080515848128051007, −5.86046716952307487820979617410, −5.44602784780516627690753626347, −4.78350469315759366100286727218, −4.09093854943951626249159972732, −2.91419684647009798125043300247, −2.58262744299758303183220447734, −1.28986792446261348818843590368, 0,
1.28986792446261348818843590368, 2.58262744299758303183220447734, 2.91419684647009798125043300247, 4.09093854943951626249159972732, 4.78350469315759366100286727218, 5.44602784780516627690753626347, 5.86046716952307487820979617410, 6.69934542100080515848128051007, 7.53154645365425419861079550746
