| L(s) = 1 | + 2.56·3-s − 2·5-s + 3.56·9-s + 5.12·11-s − 4.56·13-s − 5.12·15-s + 3.12·17-s − 5.12·19-s − 23-s − 25-s + 1.43·27-s − 0.561·29-s + 6.56·31-s + 13.1·33-s − 8.24·37-s − 11.6·39-s − 10.8·41-s − 8·43-s − 7.12·45-s − 11.6·47-s + 8·51-s + 2·53-s − 10.2·55-s − 13.1·57-s + 6.24·59-s − 12.2·61-s + 9.12·65-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.894·5-s + 1.18·9-s + 1.54·11-s − 1.26·13-s − 1.32·15-s + 0.757·17-s − 1.17·19-s − 0.208·23-s − 0.200·25-s + 0.276·27-s − 0.104·29-s + 1.17·31-s + 2.28·33-s − 1.35·37-s − 1.87·39-s − 1.68·41-s − 1.21·43-s − 1.06·45-s − 1.70·47-s + 1.12·51-s + 0.274·53-s − 1.38·55-s − 1.73·57-s + 0.813·59-s − 1.56·61-s + 1.13·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 2.31T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−7.56330079341282702145503201260, −6.86289216192958965420015074917, −6.33619493017487252300984219655, −5.02459685222775273473865310772, −4.40745828546534688848830949197, −3.57110675568849706855541784586, −3.32673412173474320840442251589, −2.21852514605938894889584562972, −1.52876411243811213366519408408, 0,
1.52876411243811213366519408408, 2.21852514605938894889584562972, 3.32673412173474320840442251589, 3.57110675568849706855541784586, 4.40745828546534688848830949197, 5.02459685222775273473865310772, 6.33619493017487252300984219655, 6.86289216192958965420015074917, 7.56330079341282702145503201260
