| L(s) = 1 | + 3-s + 2.64·5-s + 9-s + 11-s − 3.86·13-s + 2.64·15-s + 4.11·17-s + 3.53·19-s + 4.77·23-s + 2.01·25-s + 27-s + 1.99·29-s + 9.43·31-s + 33-s + 1.76·37-s − 3.86·39-s − 1.72·41-s − 1.16·43-s + 2.64·45-s − 12.1·47-s + 4.11·51-s − 5.70·53-s + 2.64·55-s + 3.53·57-s + 0.282·59-s + 2.17·61-s − 10.2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.18·5-s + 0.333·9-s + 0.301·11-s − 1.07·13-s + 0.683·15-s + 0.997·17-s + 0.811·19-s + 0.995·23-s + 0.403·25-s + 0.192·27-s + 0.369·29-s + 1.69·31-s + 0.174·33-s + 0.289·37-s − 0.619·39-s − 0.268·41-s − 0.178·43-s + 0.394·45-s − 1.77·47-s + 0.576·51-s − 0.783·53-s + 0.357·55-s + 0.468·57-s + 0.0368·59-s + 0.277·61-s − 1.27·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.544637970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.544637970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 31 | \( 1 - 9.43T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.70T + 53T^{2} \) |
| 59 | \( 1 - 0.282T + 59T^{2} \) |
| 61 | \( 1 - 2.17T + 61T^{2} \) |
| 67 | \( 1 - 6.36T + 67T^{2} \) |
| 71 | \( 1 - 3.91T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.037318300319160579961542987800, −7.31639814036192417081711344490, −6.63754519442428128355404081993, −5.91384841011162269542892323245, −5.10012889557665389551701655991, −4.59000736472241563093902366173, −3.30239422029313121888997155306, −2.80788466209697040264362141474, −1.87446274316742623249990177634, −1.00478758749422612332845433251,
1.00478758749422612332845433251, 1.87446274316742623249990177634, 2.80788466209697040264362141474, 3.30239422029313121888997155306, 4.59000736472241563093902366173, 5.10012889557665389551701655991, 5.91384841011162269542892323245, 6.63754519442428128355404081993, 7.31639814036192417081711344490, 8.037318300319160579961542987800
