| L(s) = 1 | − 2-s + 4-s + 4.25·5-s + 2.10·7-s − 8-s − 4.25·10-s − 4.67·11-s + 0.547·13-s − 2.10·14-s + 16-s − 0.524·17-s − 8.11·19-s + 4.25·20-s + 4.67·22-s + 6.00·23-s + 13.0·25-s − 0.547·26-s + 2.10·28-s + 8.07·29-s + 4.87·31-s − 32-s + 0.524·34-s + 8.94·35-s − 10.1·37-s + 8.11·38-s − 4.25·40-s − 2.58·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.90·5-s + 0.795·7-s − 0.353·8-s − 1.34·10-s − 1.41·11-s + 0.151·13-s − 0.562·14-s + 0.250·16-s − 0.127·17-s − 1.86·19-s + 0.951·20-s + 0.997·22-s + 1.25·23-s + 2.61·25-s − 0.107·26-s + 0.397·28-s + 1.49·29-s + 0.875·31-s − 0.176·32-s + 0.0900·34-s + 1.51·35-s − 1.66·37-s + 1.31·38-s − 0.672·40-s − 0.404·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.267344872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267344872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 4.25T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 0.547T + 13T^{2} \) |
| 17 | \( 1 + 0.524T + 17T^{2} \) |
| 19 | \( 1 + 8.11T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 0.544T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 0.0374T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 8.91T + 97T^{2} \) |
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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
−8.152393866736439404044304108023, −6.96809799879540151607744424414, −6.57490643296454960794291160433, −5.80474568593609851454156051247, −5.09524482567847079089829300408, −4.65880034204712042903107670677, −3.08080281593125515050954488441, −2.32848667417958022224803693624, −1.88888537209609726301578358831, −0.833309187124058500429288039362,
0.833309187124058500429288039362, 1.88888537209609726301578358831, 2.32848667417958022224803693624, 3.08080281593125515050954488441, 4.65880034204712042903107670677, 5.09524482567847079089829300408, 5.80474568593609851454156051247, 6.57490643296454960794291160433, 6.96809799879540151607744424414, 8.152393866736439404044304108023
