| L(s) = 1 | − 3.30·3-s − 4.30·7-s + 7.90·9-s − 11-s − 5.60·13-s + 0.697·17-s + 19-s + 14.2·21-s + 6.90·23-s − 16.2·27-s − 5.30·29-s + 5.60·31-s + 3.30·33-s + 0.394·37-s + 18.5·39-s − 6.21·41-s + 7.21·43-s + 1.60·47-s + 11.5·49-s − 2.30·51-s + 11.5·53-s − 3.30·57-s − 1.60·59-s − 0.302·61-s − 34.0·63-s − 8·67-s − 22.8·69-s + ⋯ |
| L(s) = 1 | − 1.90·3-s − 1.62·7-s + 2.63·9-s − 0.301·11-s − 1.55·13-s + 0.169·17-s + 0.229·19-s + 3.10·21-s + 1.44·23-s − 3.11·27-s − 0.984·29-s + 1.00·31-s + 0.574·33-s + 0.0648·37-s + 2.96·39-s − 0.970·41-s + 1.09·43-s + 0.234·47-s + 1.64·49-s − 0.322·51-s + 1.58·53-s − 0.437·57-s − 0.209·59-s − 0.0387·61-s − 4.28·63-s − 0.977·67-s − 2.74·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 - 0.697T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 0.394T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 + 0.302T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.48199159977325503947110086825, −7.12737781529924066162287917849, −6.49033531786687156533427122228, −5.78161591480581948242474395595, −5.17773799382613229435348794115, −4.50385703136353278979897587991, −3.46610161480902277505056117206, −2.42928871971558401851029105690, −0.875711662346880959343878575085, 0,
0.875711662346880959343878575085, 2.42928871971558401851029105690, 3.46610161480902277505056117206, 4.50385703136353278979897587991, 5.17773799382613229435348794115, 5.78161591480581948242474395595, 6.49033531786687156533427122228, 7.12737781529924066162287917849, 7.48199159977325503947110086825
