| L(s) = 1 | − 3.30·3-s + 0.302·7-s + 7.90·9-s − 11-s + 13-s + 2.60·17-s − 5.30·19-s − 1.00·21-s − 1.69·23-s − 5·25-s − 16.2·27-s − 5.21·29-s + 4.60·31-s + 3.30·33-s − 9.21·37-s − 3.30·39-s + 1.69·41-s − 6.60·43-s − 6·47-s − 6.90·49-s − 8.60·51-s − 4.69·53-s + 17.5·57-s + 14.6·59-s + 9.81·61-s + 2.39·63-s − 1.39·67-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.114·7-s + 2.63·9-s − 0.301·11-s + 0.277·13-s + 0.631·17-s − 1.21·19-s − 0.218·21-s − 0.353·23-s − 25-s − 3.11·27-s − 0.967·29-s + 0.827·31-s + 0.574·33-s − 1.51·37-s − 0.528·39-s + 0.265·41-s − 1.00·43-s − 0.875·47-s − 0.986·49-s − 1.20·51-s − 0.645·53-s + 2.31·57-s + 1.90·59-s + 1.25·61-s + 0.301·63-s − 0.170·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 9.51T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2.78T + 97T^{2} \) |
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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
−10.29280159214630214991780371328, −9.960376760257235278547880397631, −8.440624938938048525739059094156, −7.34300056192451868650647524984, −6.43669495214291989874544335551, −5.71170956256044572790849033422, −4.88210144647393290206038105793, −3.84891956337405485658655067063, −1.68622911624315320575901149133, 0,
1.68622911624315320575901149133, 3.84891956337405485658655067063, 4.88210144647393290206038105793, 5.71170956256044572790849033422, 6.43669495214291989874544335551, 7.34300056192451868650647524984, 8.440624938938048525739059094156, 9.960376760257235278547880397631, 10.29280159214630214991780371328
