| L(s) = 1 | + 2-s − 3-s + 4-s − 3.16·5-s − 6-s − 7-s + 8-s + 9-s − 3.16·10-s − 3.60·11-s − 12-s + 3.59·13-s − 14-s + 3.16·15-s + 16-s − 1.67·17-s + 18-s + 1.98·19-s − 3.16·20-s + 21-s − 3.60·22-s − 3.77·23-s − 24-s + 5.03·25-s + 3.59·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.41·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.00·10-s − 1.08·11-s − 0.288·12-s + 0.995·13-s − 0.267·14-s + 0.818·15-s + 0.250·16-s − 0.406·17-s + 0.235·18-s + 0.454·19-s − 0.708·20-s + 0.218·21-s − 0.768·22-s − 0.787·23-s − 0.204·24-s + 1.00·25-s + 0.704·26-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 | \( 1 + 3.16T + 5T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + 5.64T + 53T^{2} \) |
| 59 | \( 1 - 5.72T + 59T^{2} \) |
| 61 | \( 1 + 0.184T + 61T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.42826345111735590052845464255, −6.67741225329691304504187082960, −6.10522548671141803718324504427, −5.33051547048304413221631235459, −4.54857676420059613838354001878, −4.04495118046028822095847272443, −3.26557930562014363657348778305, −2.53094439939172643005559952628, −1.09349561061887244761872076851, 0,
1.09349561061887244761872076851, 2.53094439939172643005559952628, 3.26557930562014363657348778305, 4.04495118046028822095847272443, 4.54857676420059613838354001878, 5.33051547048304413221631235459, 6.10522548671141803718324504427, 6.67741225329691304504187082960, 7.42826345111735590052845464255
