| L(s) = 1 | − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s − 6·29-s + 8·31-s − 6·37-s + 6·41-s − 4·43-s − 7·49-s − 2·53-s − 4·59-s − 2·61-s + 4·67-s − 8·71-s − 10·73-s − 8·79-s − 4·83-s + 6·89-s − 2·97-s + 18·101-s − 16·103-s − 12·107-s − 2·109-s + 18·113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.203·97-s + 1.79·101-s − 1.57·103-s − 1.16·107-s − 0.191·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.730633394111998328725718228031, −8.095549656698690551636863836910, −7.47308398754126662115910632702, −6.33548194788761977078999997603, −5.74245729062321480161908713186, −4.76184067616314722575220108003, −3.84771560491769769437106298900, −2.79864903411763483594224581992, −1.72979115909686153918413689995, 0,
1.72979115909686153918413689995, 2.79864903411763483594224581992, 3.84771560491769769437106298900, 4.76184067616314722575220108003, 5.74245729062321480161908713186, 6.33548194788761977078999997603, 7.47308398754126662115910632702, 8.095549656698690551636863836910, 8.730633394111998328725718228031
