L(s) = 1 | − 2-s + 0.517·3-s + 4-s + 1.86·5-s − 0.517·6-s − 8-s − 2.73·9-s − 1.86·10-s − 1.34·11-s + 0.517·12-s + 5.21·13-s + 0.965·15-s + 16-s − 4.51·17-s + 2.73·18-s + 3.21·19-s + 1.86·20-s + 1.34·22-s − 6.18·23-s − 0.517·24-s − 1.51·25-s − 5.21·26-s − 2.96·27-s − 8.11·29-s − 0.965·30-s + 0.697·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.298·3-s + 0.5·4-s + 0.834·5-s − 0.211·6-s − 0.353·8-s − 0.910·9-s − 0.590·10-s − 0.406·11-s + 0.149·12-s + 1.44·13-s + 0.249·15-s + 0.250·16-s − 1.09·17-s + 0.644·18-s + 0.737·19-s + 0.417·20-s + 0.287·22-s − 1.28·23-s − 0.105·24-s − 0.303·25-s − 1.02·26-s − 0.570·27-s − 1.50·29-s − 0.176·30-s + 0.125·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 0.517T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 5.21T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 - 0.697T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 + 6.38T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 9.77T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.179188042935862823958957003911, −7.61830181755924330496252583109, −6.48190294119916944133632486471, −5.98482338716255638444618803658, −5.40762285656300425126810122847, −4.10568654717014208921871639053, −3.19645369105024131177586290304, −2.28430519007002257210623014372, −1.53151697054342749762635798349, 0,
1.53151697054342749762635798349, 2.28430519007002257210623014372, 3.19645369105024131177586290304, 4.10568654717014208921871639053, 5.40762285656300425126810122847, 5.98482338716255638444618803658, 6.48190294119916944133632486471, 7.61830181755924330496252583109, 8.179188042935862823958957003911