| L(s) = 1 | + 3-s + 2.90·5-s + 0.407·7-s + 9-s + 2.69·11-s + 5.88·13-s + 2.90·15-s + 1.95·17-s + 4.30·19-s + 0.407·21-s + 0.431·23-s + 3.46·25-s + 27-s + 4.98·29-s − 9.11·31-s + 2.69·33-s + 1.18·35-s − 37-s + 5.88·39-s − 7.25·41-s − 4.62·43-s + 2.90·45-s − 5.68·47-s − 6.83·49-s + 1.95·51-s − 7.52·53-s + 7.84·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30·5-s + 0.153·7-s + 0.333·9-s + 0.813·11-s + 1.63·13-s + 0.751·15-s + 0.475·17-s + 0.987·19-s + 0.0889·21-s + 0.0899·23-s + 0.692·25-s + 0.192·27-s + 0.926·29-s − 1.63·31-s + 0.469·33-s + 0.200·35-s − 0.164·37-s + 0.941·39-s − 1.13·41-s − 0.704·43-s + 0.433·45-s − 0.828·47-s − 0.976·49-s + 0.274·51-s − 1.03·53-s + 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.705141565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.705141565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 0.407T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 - 1.95T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 - 0.431T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 + 9.11T + 31T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 + 4.62T + 43T^{2} \) |
| 47 | \( 1 + 5.68T + 47T^{2} \) |
| 53 | \( 1 + 7.52T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 7.81T + 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 - 1.28T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.664233983226454359515572380276, −7.993111466076396360890201457110, −6.96029204030004067696043604793, −6.32012667384638859910024165348, −5.65247076485458992235805446337, −4.83889222814005728129891011786, −3.65198469760573089814656877932, −3.12634711968562177319889809285, −1.76657952762139358369556537258, −1.32732346097394884494709339288,
1.32732346097394884494709339288, 1.76657952762139358369556537258, 3.12634711968562177319889809285, 3.65198469760573089814656877932, 4.83889222814005728129891011786, 5.65247076485458992235805446337, 6.32012667384638859910024165348, 6.96029204030004067696043604793, 7.993111466076396360890201457110, 8.664233983226454359515572380276
