| L(s) = 1 | − 0.806·3-s − 5-s + 4.15·7-s − 2.35·9-s − 1.19·11-s − 5.35·13-s + 0.806·15-s + 3.76·17-s − 6.15·19-s − 3.35·21-s − 0.806·23-s + 25-s + 4.31·27-s + 29-s − 4.54·31-s + 0.962·33-s − 4.15·35-s + 2.15·37-s + 4.31·39-s + 1.03·41-s − 6.41·43-s + 2.35·45-s − 6.73·47-s + 10.2·49-s − 3.03·51-s − 12.8·53-s + 1.19·55-s + ⋯ |
| L(s) = 1 | − 0.465·3-s − 0.447·5-s + 1.57·7-s − 0.783·9-s − 0.359·11-s − 1.48·13-s + 0.208·15-s + 0.913·17-s − 1.41·19-s − 0.731·21-s − 0.168·23-s + 0.200·25-s + 0.829·27-s + 0.185·29-s − 0.816·31-s + 0.167·33-s − 0.702·35-s + 0.354·37-s + 0.690·39-s + 0.162·41-s − 0.978·43-s + 0.350·45-s − 0.981·47-s + 1.46·49-s − 0.425·51-s − 1.77·53-s + 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 + 0.806T + 23T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 6.57T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−9.361573484200902235919780570401, −8.237084907361474488113598693044, −7.962273388700895410542777856644, −6.98593514770264025033792752164, −5.81345519715399259928444174118, −5.00422410675747259189941423415, −4.44525659583463715283573419450, −2.96102578644912834313670958732, −1.78836548999288821023160436933, 0,
1.78836548999288821023160436933, 2.96102578644912834313670958732, 4.44525659583463715283573419450, 5.00422410675747259189941423415, 5.81345519715399259928444174118, 6.98593514770264025033792752164, 7.962273388700895410542777856644, 8.237084907361474488113598693044, 9.361573484200902235919780570401
