| L(s) = 1 | − 2·7-s + 34·11-s + 68·13-s − 38·17-s + 4·19-s + 152·23-s + 46·29-s − 260·31-s + 312·37-s − 48·41-s + 200·43-s + 104·47-s − 339·49-s − 414·53-s + 2·59-s − 38·61-s + 244·67-s − 708·71-s + 378·73-s − 68·77-s − 852·79-s + 844·83-s + 1.38e3·89-s − 136·91-s − 514·97-s + 702·101-s − 898·103-s + ⋯ |
| L(s) = 1 | − 0.107·7-s + 0.931·11-s + 1.45·13-s − 0.542·17-s + 0.0482·19-s + 1.37·23-s + 0.294·29-s − 1.50·31-s + 1.38·37-s − 0.182·41-s + 0.709·43-s + 0.322·47-s − 0.988·49-s − 1.07·53-s + 0.00441·59-s − 0.0797·61-s + 0.444·67-s − 1.18·71-s + 0.606·73-s − 0.100·77-s − 1.21·79-s + 1.11·83-s + 1.64·89-s − 0.156·91-s − 0.538·97-s + 0.691·101-s − 0.859·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.558029174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558029174\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 46 T + p^{3} T^{2} \) |
| 31 | \( 1 + 260 T + p^{3} T^{2} \) |
| 37 | \( 1 - 312 T + p^{3} T^{2} \) |
| 41 | \( 1 + 48 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 - 104 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 2 T + p^{3} T^{2} \) |
| 61 | \( 1 + 38 T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 708 T + p^{3} T^{2} \) |
| 73 | \( 1 - 378 T + p^{3} T^{2} \) |
| 79 | \( 1 + 852 T + p^{3} T^{2} \) |
| 83 | \( 1 - 844 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1380 T + p^{3} T^{2} \) |
| 97 | \( 1 + 514 T + p^{3} T^{2} \) |
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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
−9.058393033446954158002746326849, −8.230511500297062422760279405888, −7.25862509191813268101227697770, −6.47343385796858788826772794614, −5.85934141867093915964848343019, −4.74310708823587236638044154386, −3.86954713032924663129253262293, −3.07292510990773006036643817888, −1.73213950548905009883334713324, −0.793724340054898864456204462597,
0.793724340054898864456204462597, 1.73213950548905009883334713324, 3.07292510990773006036643817888, 3.86954713032924663129253262293, 4.74310708823587236638044154386, 5.85934141867093915964848343019, 6.47343385796858788826772794614, 7.25862509191813268101227697770, 8.230511500297062422760279405888, 9.058393033446954158002746326849
