| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 4·13-s + 2·17-s − 4·19-s + 21-s − 3·23-s − 27-s − 29-s − 6·31-s + 33-s − 3·37-s − 4·39-s + 5·43-s + 4·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s − 4·59-s − 4·61-s − 63-s − 5·67-s + 3·69-s − 71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s − 0.185·29-s − 1.07·31-s + 0.174·33-s − 0.493·37-s − 0.640·39-s + 0.762·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.512·61-s − 0.125·63-s − 0.610·67-s + 0.361·69-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.007423754393233233223064879668, −7.26652767279762748754883051495, −6.43181558544078211324804244923, −5.88194866135640674259632252136, −5.21266362281290383661016023931, −4.13751966025694296501822914038, −3.57970238326339769337297315250, −2.39983654278846278018108757064, −1.30700570439287422397215342298, 0,
1.30700570439287422397215342298, 2.39983654278846278018108757064, 3.57970238326339769337297315250, 4.13751966025694296501822914038, 5.21266362281290383661016023931, 5.88194866135640674259632252136, 6.43181558544078211324804244923, 7.26652767279762748754883051495, 8.007423754393233233223064879668
