| L(s) = 1 | − 3-s + 9-s − 13-s + 2·19-s + 23-s − 27-s + 2·29-s − 8·31-s + 3·37-s + 39-s − 12·41-s − 8·43-s − 9·47-s − 7·49-s + 6·53-s − 2·57-s − 15·59-s + 5·61-s − 2·67-s − 69-s − 3·71-s − 15·73-s − 8·79-s + 81-s − 3·83-s − 2·87-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.277·13-s + 0.458·19-s + 0.208·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.493·37-s + 0.160·39-s − 1.87·41-s − 1.21·43-s − 1.31·47-s − 49-s + 0.824·53-s − 0.264·57-s − 1.95·59-s + 0.640·61-s − 0.244·67-s − 0.120·69-s − 0.356·71-s − 1.75·73-s − 0.900·79-s + 1/9·81-s − 0.329·83-s − 0.214·87-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2153184093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2153184093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.16981082963311, −12.78896914434333, −12.07324133991850, −11.88149295244094, −11.25814141338015, −11.00179274020796, −10.26763074186807, −9.938598225125158, −9.556218653191976, −8.815379535262163, −8.496343349976653, −7.804675423619679, −7.381096983952602, −6.759162627855976, −6.530276908263494, −5.667196087181256, −5.461306721506751, −4.714524029241762, −4.482261709018275, −3.532286842204768, −3.237292642426471, −2.498890123919198, −1.570975559551283, −1.386713209795239, −0.1367431226893115,
0.1367431226893115, 1.386713209795239, 1.570975559551283, 2.498890123919198, 3.237292642426471, 3.532286842204768, 4.482261709018275, 4.714524029241762, 5.461306721506751, 5.667196087181256, 6.530276908263494, 6.759162627855976, 7.381096983952602, 7.804675423619679, 8.496343349976653, 8.815379535262163, 9.556218653191976, 9.938598225125158, 10.26763074186807, 11.00179274020796, 11.25814141338015, 11.88149295244094, 12.07324133991850, 12.78896914434333, 13.16981082963311
