| L(s) = 1 | − 3-s − 2.94·5-s + 4.65·7-s + 9-s − 5.56·11-s − 0.393·13-s + 2.94·15-s − 0.257·17-s + 2.11·19-s − 4.65·21-s + 5.38·23-s + 3.68·25-s − 27-s − 0.101·29-s − 3.82·31-s + 5.56·33-s − 13.7·35-s − 1.36·37-s + 0.393·39-s − 5.79·41-s + 7.94·43-s − 2.94·45-s + 5.64·47-s + 14.6·49-s + 0.257·51-s − 4.37·53-s + 16.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.31·5-s + 1.75·7-s + 0.333·9-s − 1.67·11-s − 0.109·13-s + 0.761·15-s − 0.0625·17-s + 0.484·19-s − 1.01·21-s + 1.12·23-s + 0.737·25-s − 0.192·27-s − 0.0187·29-s − 0.686·31-s + 0.968·33-s − 2.31·35-s − 0.224·37-s + 0.0630·39-s − 0.904·41-s + 1.21·43-s − 0.439·45-s + 0.822·47-s + 2.09·49-s + 0.0361·51-s − 0.600·53-s + 2.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108516739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108516739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.94T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 0.393T + 13T^{2} \) |
| 17 | \( 1 + 0.257T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 - 5.64T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 7.40T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−8.892282284019549059439077001221, −8.202484049835363278760649020942, −7.52724647273812489613387648047, −7.24420717814293494494393304385, −5.68904572126130629872067437819, −4.99491889566067123757522236055, −4.53853742570402142956240018037, −3.40090986754436407475533162254, −2.14267024985015277405685276084, −0.72137093109956444727320594778,
0.72137093109956444727320594778, 2.14267024985015277405685276084, 3.40090986754436407475533162254, 4.53853742570402142956240018037, 4.99491889566067123757522236055, 5.68904572126130629872067437819, 7.24420717814293494494393304385, 7.52724647273812489613387648047, 8.202484049835363278760649020942, 8.892282284019549059439077001221
