| L(s) = 1 | + 2-s − 3-s + 4-s + 2.03·5-s − 6-s − 2.78·7-s + 8-s + 9-s + 2.03·10-s + 2.25·11-s − 12-s − 0.0372·13-s − 2.78·14-s − 2.03·15-s + 16-s − 17-s + 18-s − 1.18·19-s + 2.03·20-s + 2.78·21-s + 2.25·22-s − 5.57·23-s − 24-s − 0.871·25-s − 0.0372·26-s − 27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.908·5-s − 0.408·6-s − 1.05·7-s + 0.353·8-s + 0.333·9-s + 0.642·10-s + 0.679·11-s − 0.288·12-s − 0.0103·13-s − 0.743·14-s − 0.524·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.270·19-s + 0.454·20-s + 0.606·21-s + 0.480·22-s − 1.16·23-s − 0.204·24-s − 0.174·25-s − 0.00730·26-s − 0.192·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.795035781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.795035781\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 + 0.0372T + 13T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 - 9.92T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 - 0.151T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 61 | \( 1 + 1.98T + 61T^{2} \) |
| 67 | \( 1 - 0.598T + 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
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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
−7.911160002083659247124862545276, −7.03130422147611099621307649865, −6.29871303711254853784253053413, −6.10786629824207810290644357299, −5.38558001690659468180555500095, −4.36726785622280410786692196411, −3.84909512571377985576138502148, −2.75638121555739264364993838759, −2.03106197751387998016086735521, −0.810234281344195161223751030555,
0.810234281344195161223751030555, 2.03106197751387998016086735521, 2.75638121555739264364993838759, 3.84909512571377985576138502148, 4.36726785622280410786692196411, 5.38558001690659468180555500095, 6.10786629824207810290644357299, 6.29871303711254853784253053413, 7.03130422147611099621307649865, 7.911160002083659247124862545276
