L(s) = 1 | − 1.70·3-s + 5-s − 1.03·7-s − 0.0914·9-s + 1.24·11-s − 4.47·13-s − 1.70·15-s + 2.95·17-s + 0.264·19-s + 1.77·21-s − 1.16·23-s + 25-s + 5.27·27-s − 2.37·29-s − 3.78·31-s − 2.12·33-s − 1.03·35-s + 10.8·37-s + 7.62·39-s − 3.50·41-s − 5.37·43-s − 0.0914·45-s − 7.48·47-s − 5.92·49-s − 5.04·51-s + 1.25·53-s + 1.24·55-s + ⋯ |
L(s) = 1 | − 0.984·3-s + 0.447·5-s − 0.392·7-s − 0.0304·9-s + 0.376·11-s − 1.24·13-s − 0.440·15-s + 0.717·17-s + 0.0607·19-s + 0.386·21-s − 0.242·23-s + 0.200·25-s + 1.01·27-s − 0.440·29-s − 0.679·31-s − 0.370·33-s − 0.175·35-s + 1.78·37-s + 1.22·39-s − 0.547·41-s − 0.819·43-s − 0.0136·45-s − 1.09·47-s − 0.845·49-s − 0.706·51-s + 0.173·53-s + 0.168·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9449184161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9449184161\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.95T + 17T^{2} \) |
| 19 | \( 1 - 0.264T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 3.50T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 0.329T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 - 9.25T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−7.68848293857400755674052174391, −7.00838149806762462112769955780, −6.33730086413765532324517171059, −5.77505304005798257010669146540, −5.15142266825322492047118227868, −4.53216852850164772155388721189, −3.46850323219259280015654938996, −2.67793563853237510802078682615, −1.66967182320403057176856939022, −0.50231741025325767934148112425,
0.50231741025325767934148112425, 1.66967182320403057176856939022, 2.67793563853237510802078682615, 3.46850323219259280015654938996, 4.53216852850164772155388721189, 5.15142266825322492047118227868, 5.77505304005798257010669146540, 6.33730086413765532324517171059, 7.00838149806762462112769955780, 7.68848293857400755674052174391