L(s) = 1 | − 0.693·2-s + 3-s − 1.51·4-s − 5-s − 0.693·6-s + 5.04·7-s + 2.44·8-s + 9-s + 0.693·10-s − 4.10·11-s − 1.51·12-s + 0.261·13-s − 3.49·14-s − 15-s + 1.34·16-s − 2.76·17-s − 0.693·18-s + 3.27·19-s + 1.51·20-s + 5.04·21-s + 2.84·22-s + 1.69·23-s + 2.44·24-s + 25-s − 0.181·26-s + 27-s − 7.66·28-s + ⋯ |
L(s) = 1 | − 0.490·2-s + 0.577·3-s − 0.759·4-s − 0.447·5-s − 0.283·6-s + 1.90·7-s + 0.862·8-s + 0.333·9-s + 0.219·10-s − 1.23·11-s − 0.438·12-s + 0.0724·13-s − 0.935·14-s − 0.258·15-s + 0.336·16-s − 0.670·17-s − 0.163·18-s + 0.750·19-s + 0.339·20-s + 1.10·21-s + 0.607·22-s + 0.352·23-s + 0.498·24-s + 0.200·25-s − 0.0355·26-s + 0.192·27-s − 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658124332\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658124332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 0.693T + 2T^{2} \) |
| 7 | \( 1 - 5.04T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 - 0.261T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 + 9.68T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−8.155911829210400788191406472833, −7.51241427439474833889167802612, −7.29987140260874091169661038360, −5.59255541105859488625620606992, −5.15390959221580656763072999302, −4.40947653046406228370073375077, −3.84778901158507127161159628795, −2.60411677361979429404023755165, −1.76546591426697395127413514309, −0.73980899786166803148976668473,
0.73980899786166803148976668473, 1.76546591426697395127413514309, 2.60411677361979429404023755165, 3.84778901158507127161159628795, 4.40947653046406228370073375077, 5.15390959221580656763072999302, 5.59255541105859488625620606992, 7.29987140260874091169661038360, 7.51241427439474833889167802612, 8.155911829210400788191406472833