| L(s) = 1 | − 2.28·3-s + 1.50·5-s + 2.22·9-s + 2.10·11-s + 3.79·13-s − 3.43·15-s − 2.43·17-s − 19-s + 3.12·23-s − 2.74·25-s + 1.77·27-s − 8.00·29-s + 8.70·31-s − 4.80·33-s + 3.59·37-s − 8.68·39-s + 0.873·41-s + 5.89·43-s + 3.34·45-s + 4.76·47-s + 5.56·51-s − 11.4·53-s + 3.15·55-s + 2.28·57-s + 1.76·59-s + 12.6·61-s + 5.70·65-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 0.671·5-s + 0.741·9-s + 0.634·11-s + 1.05·13-s − 0.886·15-s − 0.590·17-s − 0.229·19-s + 0.651·23-s − 0.548·25-s + 0.341·27-s − 1.48·29-s + 1.56·31-s − 0.836·33-s + 0.591·37-s − 1.39·39-s + 0.136·41-s + 0.898·43-s + 0.498·45-s + 0.695·47-s + 0.779·51-s − 1.56·53-s + 0.426·55-s + 0.302·57-s + 0.229·59-s + 1.62·61-s + 0.707·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.463452617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463452617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.28T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 - 0.873T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 2.84T + 97T^{2} \) |
| 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
−7.83473244283459634431063036256, −6.85946449305934148901324929319, −6.35084021967390760788047408751, −5.89886494791926528474643818372, −5.27659326122238505351731177661, −4.43249982950517200909708268389, −3.75691997088265340831970380138, −2.57894434228785023446464824389, −1.55389604807115672466591522309, −0.68998048756600089271000188327,
0.68998048756600089271000188327, 1.55389604807115672466591522309, 2.57894434228785023446464824389, 3.75691997088265340831970380138, 4.43249982950517200909708268389, 5.27659326122238505351731177661, 5.89886494791926528474643818372, 6.35084021967390760788047408751, 6.85946449305934148901324929319, 7.83473244283459634431063036256
