| L(s) = 1 | − 1.37·2-s + 3-s − 0.117·4-s + 2.69·5-s − 1.37·6-s − 7-s + 2.90·8-s + 9-s − 3.69·10-s + 2.84·11-s − 0.117·12-s − 0.726·13-s + 1.37·14-s + 2.69·15-s − 3.75·16-s + 2.51·17-s − 1.37·18-s + 6.41·19-s − 0.316·20-s − 21-s − 3.89·22-s + 1.25·23-s + 2.90·24-s + 2.26·25-s + 0.996·26-s + 27-s + 0.117·28-s + ⋯ |
| L(s) = 1 | − 0.970·2-s + 0.577·3-s − 0.0587·4-s + 1.20·5-s − 0.560·6-s − 0.377·7-s + 1.02·8-s + 0.333·9-s − 1.16·10-s + 0.856·11-s − 0.0339·12-s − 0.201·13-s + 0.366·14-s + 0.695·15-s − 0.937·16-s + 0.610·17-s − 0.323·18-s + 1.47·19-s − 0.0707·20-s − 0.218·21-s − 0.831·22-s + 0.262·23-s + 0.593·24-s + 0.452·25-s + 0.195·26-s + 0.192·27-s + 0.0221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107641823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107641823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 0.726T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 - 0.121T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 0.956T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 8.77T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−7.79636335609958595126076843657, −7.43751899459448663992635279967, −6.55404758675912043364165098654, −5.88227768679582784620887505414, −5.07062838440732146924863042267, −4.26788626347397044850834135600, −3.32073647185574198472863011895, −2.50660064830111837187804208481, −1.51557994378136236553859444731, −0.910121041423657043483552043332,
0.910121041423657043483552043332, 1.51557994378136236553859444731, 2.50660064830111837187804208481, 3.32073647185574198472863011895, 4.26788626347397044850834135600, 5.07062838440732146924863042267, 5.88227768679582784620887505414, 6.55404758675912043364165098654, 7.43751899459448663992635279967, 7.79636335609958595126076843657
