| L(s) = 1 | + 2.23·3-s + 1.23·5-s − 3.23·7-s + 2.00·9-s + 5.23·11-s + 3·13-s + 2.76·15-s + 0.763·17-s + 2·19-s − 7.23·21-s − 23-s − 3.47·25-s − 2.23·27-s − 3·29-s − 6.70·31-s + 11.7·33-s − 4.00·35-s − 1.23·37-s + 6.70·39-s − 3.47·41-s + 2.47·45-s + 2.23·47-s + 3.47·49-s + 1.70·51-s + 0.472·53-s + 6.47·55-s + 4.47·57-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 0.552·5-s − 1.22·7-s + 0.666·9-s + 1.57·11-s + 0.832·13-s + 0.713·15-s + 0.185·17-s + 0.458·19-s − 1.57·21-s − 0.208·23-s − 0.694·25-s − 0.430·27-s − 0.557·29-s − 1.20·31-s + 2.03·33-s − 0.676·35-s − 0.203·37-s + 1.07·39-s − 0.542·41-s + 0.368·45-s + 0.326·47-s + 0.496·49-s + 0.239·51-s + 0.0648·53-s + 0.872·55-s + 0.592·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060558756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060558756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 2.76T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 6.52T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−11.44420022844019217459730267824, −10.14352922351397147710915189939, −9.199727331679352674644043276800, −9.082813866517354113111444850445, −7.74343865236245248842713373798, −6.63569228682367228543134706324, −5.81728743237325468371592233222, −3.87133324097187052973248182514, −3.26964018939303481314596199283, −1.77386441832074138732796153972,
1.77386441832074138732796153972, 3.26964018939303481314596199283, 3.87133324097187052973248182514, 5.81728743237325468371592233222, 6.63569228682367228543134706324, 7.74343865236245248842713373798, 9.082813866517354113111444850445, 9.199727331679352674644043276800, 10.14352922351397147710915189939, 11.44420022844019217459730267824
