| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 2·13-s − 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 21-s + 23-s − 24-s − 25-s + 2·26-s + 27-s + 28-s + 2·29-s + 2·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.81209434081805, −13.25873768337327, −12.98084301343468, −12.24759328744777, −11.70509437698442, −11.44335390182668, −11.01650881175464, −10.25277265570293, −9.977408120145281, −9.341990897231104, −8.842353841106894, −8.311797069011382, −8.119098650459218, −7.402094906355796, −7.145670516051474, −6.445869143784325, −6.036856558623885, −4.945098666407316, −4.704593248960706, −4.052289625490557, −3.450167916347845, −2.773015813406504, −2.249138921346508, −1.627697903424404, −0.7465298466096718, 0,
0.7465298466096718, 1.627697903424404, 2.249138921346508, 2.773015813406504, 3.450167916347845, 4.052289625490557, 4.704593248960706, 4.945098666407316, 6.036856558623885, 6.445869143784325, 7.145670516051474, 7.402094906355796, 8.119098650459218, 8.311797069011382, 8.842353841106894, 9.341990897231104, 9.977408120145281, 10.25277265570293, 11.01650881175464, 11.44335390182668, 11.70509437698442, 12.24759328744777, 12.98084301343468, 13.25873768337327, 13.81209434081805
