| L(s) = 1 | + 2.13·2-s − 3-s + 2.57·4-s − 3.64·5-s − 2.13·6-s − 4.64·7-s + 1.22·8-s + 9-s − 7.78·10-s − 0.795·11-s − 2.57·12-s − 0.469·13-s − 9.92·14-s + 3.64·15-s − 2.52·16-s − 17-s + 2.13·18-s + 3.75·19-s − 9.36·20-s + 4.64·21-s − 1.70·22-s − 5.28·23-s − 1.22·24-s + 8.25·25-s − 1.00·26-s − 27-s − 11.9·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.28·4-s − 1.62·5-s − 0.872·6-s − 1.75·7-s + 0.432·8-s + 0.333·9-s − 2.46·10-s − 0.239·11-s − 0.742·12-s − 0.130·13-s − 2.65·14-s + 0.940·15-s − 0.632·16-s − 0.242·17-s + 0.503·18-s + 0.860·19-s − 2.09·20-s + 1.01·21-s − 0.362·22-s − 1.10·23-s − 0.249·24-s + 1.65·25-s − 0.196·26-s − 0.192·27-s − 2.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.123963077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123963077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 + 0.795T + 11T^{2} \) |
| 13 | \( 1 + 0.469T + 13T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.135T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 0.755T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−8.164450928465752843647631252501, −7.42605944151744871889081944054, −6.61609338943054481439798970976, −6.30057701116770547123552490962, −5.30654122043749210076405537712, −4.61154384623744366702021319633, −3.73694685144579931650159738769, −3.45792130130296208399254999172, −2.54580783717044748219705007174, −0.46499392726948219064697004202,
0.46499392726948219064697004202, 2.54580783717044748219705007174, 3.45792130130296208399254999172, 3.73694685144579931650159738769, 4.61154384623744366702021319633, 5.30654122043749210076405537712, 6.30057701116770547123552490962, 6.61609338943054481439798970976, 7.42605944151744871889081944054, 8.164450928465752843647631252501
