| L(s) = 1 | + 2-s + 1.12·3-s + 4-s − 1.71·5-s + 1.12·6-s + 8-s − 1.73·9-s − 1.71·10-s − 3.64·11-s + 1.12·12-s + 3.79·13-s − 1.93·15-s + 16-s − 4.17·17-s − 1.73·18-s − 8.12·19-s − 1.71·20-s − 3.64·22-s − 23-s + 1.12·24-s − 2.05·25-s + 3.79·26-s − 5.33·27-s + 2.80·29-s − 1.93·30-s − 8.54·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.650·3-s + 0.5·4-s − 0.767·5-s + 0.459·6-s + 0.353·8-s − 0.576·9-s − 0.542·10-s − 1.10·11-s + 0.325·12-s + 1.05·13-s − 0.499·15-s + 0.250·16-s − 1.01·17-s − 0.407·18-s − 1.86·19-s − 0.383·20-s − 0.778·22-s − 0.208·23-s + 0.229·24-s − 0.410·25-s + 0.744·26-s − 1.02·27-s + 0.521·29-s − 0.353·30-s − 1.53·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 + 8.12T + 19T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 4.18T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 3.95T + 89T^{2} \) |
| 97 | \( 1 + 0.121T + 97T^{2} \) |
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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
−8.396897080419425824188736343564, −8.063636986538468366644751764795, −7.10322080964341959227671457177, −6.19821485634967083884985383190, −5.48595715182361411907996664351, −4.31110453197249492877019353848, −3.84014737379571252577701075650, −2.79616936729187411087471463730, −2.05003806605063785597691115905, 0,
2.05003806605063785597691115905, 2.79616936729187411087471463730, 3.84014737379571252577701075650, 4.31110453197249492877019353848, 5.48595715182361411907996664351, 6.19821485634967083884985383190, 7.10322080964341959227671457177, 8.063636986538468366644751764795, 8.396897080419425824188736343564
