| L(s) = 1 | − 2.66·2-s + 3-s + 5.12·4-s + 1.20·5-s − 2.66·6-s − 8.35·8-s + 9-s − 3.21·10-s − 1.40·11-s + 5.12·12-s − 4.24·13-s + 1.20·15-s + 12.0·16-s + 2.55·17-s − 2.66·18-s − 0.414·19-s + 6.16·20-s + 3.74·22-s − 0.127·23-s − 8.35·24-s − 3.55·25-s + 11.3·26-s + 27-s + 4.65·29-s − 3.21·30-s − 1.86·31-s − 15.4·32-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.577·3-s + 2.56·4-s + 0.537·5-s − 1.08·6-s − 2.95·8-s + 0.333·9-s − 1.01·10-s − 0.422·11-s + 1.48·12-s − 1.17·13-s + 0.310·15-s + 3.00·16-s + 0.618·17-s − 0.629·18-s − 0.0951·19-s + 1.37·20-s + 0.798·22-s − 0.0266·23-s − 1.70·24-s − 0.710·25-s + 2.22·26-s + 0.192·27-s + 0.863·29-s − 0.586·30-s − 0.334·31-s − 2.72·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 + 0.414T + 19T^{2} \) |
| 23 | \( 1 + 0.127T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 0.948T + 37T^{2} \) |
| 41 | \( 1 + 0.438T + 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 0.881T + 79T^{2} \) |
| 83 | \( 1 - 2.22T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 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
−7.67999016392171722795573893313, −7.08888755515605873150008982157, −6.32501843972397893799731639193, −5.63006985285256977391126590829, −4.66955238481822890222452627230, −3.38781560382447854884528288307, −2.58940683299046581896525989246, −2.07880957283591353408162313252, −1.17276964960624788804610163525, 0,
1.17276964960624788804610163525, 2.07880957283591353408162313252, 2.58940683299046581896525989246, 3.38781560382447854884528288307, 4.66955238481822890222452627230, 5.63006985285256977391126590829, 6.32501843972397893799731639193, 7.08888755515605873150008982157, 7.67999016392171722795573893313
