L(s) = 1 | + 2.79·3-s + 5-s + 1.22·7-s + 4.80·9-s + 3.73·11-s + 2.79·15-s + 4.95·17-s − 3.06·19-s + 3.43·21-s − 1.04·23-s + 25-s + 5.05·27-s + 9.02·29-s − 6.57·31-s + 10.4·33-s + 1.22·35-s − 0.619·37-s − 8.41·41-s − 8.50·43-s + 4.80·45-s − 5.65·47-s − 5.48·49-s + 13.8·51-s + 9.83·53-s + 3.73·55-s − 8.56·57-s + 12.1·59-s + ⋯ |
L(s) = 1 | + 1.61·3-s + 0.447·5-s + 0.464·7-s + 1.60·9-s + 1.12·11-s + 0.721·15-s + 1.20·17-s − 0.703·19-s + 0.749·21-s − 0.217·23-s + 0.200·25-s + 0.973·27-s + 1.67·29-s − 1.18·31-s + 1.81·33-s + 0.207·35-s − 0.101·37-s − 1.31·41-s − 1.29·43-s + 0.717·45-s − 0.825·47-s − 0.784·49-s + 1.93·51-s + 1.35·53-s + 0.503·55-s − 1.13·57-s + 1.58·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.353656881\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.353656881\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 17 | \( 1 - 4.95T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 9.02T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 0.619T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 9.83T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.532788081220020558518639240824, −8.192026175493968912563513191588, −7.19366566873401399286377964443, −6.61506019061828444183001471585, −5.55423265828628526665413484788, −4.57405329053351165297999293374, −3.71900385525860941604263192285, −3.07910391237843663535014692760, −2.02591443957035818811741365489, −1.34339344013668806985292165979,
1.34339344013668806985292165979, 2.02591443957035818811741365489, 3.07910391237843663535014692760, 3.71900385525860941604263192285, 4.57405329053351165297999293374, 5.55423265828628526665413484788, 6.61506019061828444183001471585, 7.19366566873401399286377964443, 8.192026175493968912563513191588, 8.532788081220020558518639240824