| L(s) = 1 | + 2.77·2-s − 3-s + 5.71·4-s − 5-s − 2.77·6-s − 2.71·7-s + 10.3·8-s + 9-s − 2.77·10-s − 2.71·11-s − 5.71·12-s + 13-s − 7.55·14-s + 15-s + 17.2·16-s − 2.83·17-s + 2.77·18-s − 3.55·19-s − 5.71·20-s + 2.71·21-s − 7.55·22-s − 4.83·23-s − 10.3·24-s + 25-s + 2.77·26-s − 27-s − 15.5·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.85·4-s − 0.447·5-s − 1.13·6-s − 1.02·7-s + 3.65·8-s + 0.333·9-s − 0.878·10-s − 0.820·11-s − 1.65·12-s + 0.277·13-s − 2.01·14-s + 0.258·15-s + 4.31·16-s − 0.688·17-s + 0.654·18-s − 0.816·19-s − 1.27·20-s + 0.593·21-s − 1.61·22-s − 1.00·23-s − 2.10·24-s + 0.200·25-s + 0.544·26-s − 0.192·27-s − 2.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.627805770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627805770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−12.63618771552804096095883981421, −11.91871267860950945745351729826, −10.92448731660374992066144017440, −10.16625732954689913815370092997, −8.000107760159811090368327265033, −6.69456209289740582008755243469, −6.14713443537015643627473165478, −4.88903343096329921589187891531, −3.90464258446346061531112922707, −2.60608140589898025960625855617,
2.60608140589898025960625855617, 3.90464258446346061531112922707, 4.88903343096329921589187891531, 6.14713443537015643627473165478, 6.69456209289740582008755243469, 8.000107760159811090368327265033, 10.16625732954689913815370092997, 10.92448731660374992066144017440, 11.91871267860950945745351729826, 12.63618771552804096095883981421
