| L(s) = 1 | − 2·11-s − 6·13-s + 17-s + 4·19-s − 4·23-s − 5·25-s − 8·31-s − 4·37-s − 6·41-s + 8·43-s + 8·47-s − 7·49-s − 10·53-s + 12·61-s + 8·67-s − 12·71-s + 2·73-s − 4·79-s − 16·83-s − 10·89-s − 18·97-s − 10·101-s + 18·107-s + 6·113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 1.66·13-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 25-s − 1.43·31-s − 0.657·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 49-s − 1.37·53-s + 1.53·61-s + 0.977·67-s − 1.42·71-s + 0.234·73-s − 0.450·79-s − 1.75·83-s − 1.05·89-s − 1.82·97-s − 0.995·101-s + 1.74·107-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.625406509757598645302901584470, −8.417457325737154058407963592246, −7.56400176257965263627932886459, −7.09070832417543820123167533646, −5.74580019409002215172587712055, −5.17682892958574806985483154514, −4.09383004007619438966415223632, −2.95072257084128076659869010441, −1.90305516483713851254985692530, 0,
1.90305516483713851254985692530, 2.95072257084128076659869010441, 4.09383004007619438966415223632, 5.17682892958574806985483154514, 5.74580019409002215172587712055, 7.09070832417543820123167533646, 7.56400176257965263627932886459, 8.417457325737154058407963592246, 9.625406509757598645302901584470
