| L(s) = 1 | + 3.56·5-s − 3.12·7-s − 1.56·11-s + 0.438·13-s − 17-s + 2.43·19-s + 2.43·23-s + 7.68·25-s − 2·29-s + 3.12·31-s − 11.1·35-s + 5.12·37-s + 8.43·41-s − 0.684·43-s − 8·47-s + 2.75·49-s − 1.12·53-s − 5.56·55-s − 11.1·59-s + 8.24·61-s + 1.56·65-s + 14.2·67-s + 11.3·73-s + 4.87·77-s + 17.3·79-s − 16·83-s − 3.56·85-s + ⋯ |
| L(s) = 1 | + 1.59·5-s − 1.18·7-s − 0.470·11-s + 0.121·13-s − 0.242·17-s + 0.559·19-s + 0.508·23-s + 1.53·25-s − 0.371·29-s + 0.560·31-s − 1.88·35-s + 0.842·37-s + 1.31·41-s − 0.104·43-s − 1.16·47-s + 0.393·49-s − 0.154·53-s − 0.749·55-s − 1.44·59-s + 1.05·61-s + 0.193·65-s + 1.74·67-s + 1.33·73-s + 0.555·77-s + 1.95·79-s − 1.75·83-s − 0.386·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.280283676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.280283676\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 + 0.684T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.335051480914696083599931849318, −7.45056434230542313208782876735, −6.52999223362521705058857657832, −6.21869150414464303531816796965, −5.45057868863599118277851776065, −4.76672073427929673205113221927, −3.53752667814212079204684377073, −2.77264162616500524570971010206, −2.05044953434936156410431843923, −0.825936137238459013008942887493,
0.825936137238459013008942887493, 2.05044953434936156410431843923, 2.77264162616500524570971010206, 3.53752667814212079204684377073, 4.76672073427929673205113221927, 5.45057868863599118277851776065, 6.21869150414464303531816796965, 6.52999223362521705058857657832, 7.45056434230542313208782876735, 8.335051480914696083599931849318
