| L(s) = 1 | + 2-s − 1.49·3-s + 4-s + 3.62·5-s − 1.49·6-s − 1.50·7-s + 8-s − 0.762·9-s + 3.62·10-s − 3.49·11-s − 1.49·12-s + 2.70·13-s − 1.50·14-s − 5.42·15-s + 16-s − 4.55·17-s − 0.762·18-s − 4.73·19-s + 3.62·20-s + 2.24·21-s − 3.49·22-s − 6.90·23-s − 1.49·24-s + 8.16·25-s + 2.70·26-s + 5.62·27-s − 1.50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.863·3-s + 0.5·4-s + 1.62·5-s − 0.610·6-s − 0.568·7-s + 0.353·8-s − 0.254·9-s + 1.14·10-s − 1.05·11-s − 0.431·12-s + 0.749·13-s − 0.401·14-s − 1.40·15-s + 0.250·16-s − 1.10·17-s − 0.179·18-s − 1.08·19-s + 0.811·20-s + 0.490·21-s − 0.745·22-s − 1.43·23-s − 0.305·24-s + 1.63·25-s + 0.529·26-s + 1.08·27-s − 0.284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 - 9.73T + 29T^{2} \) |
| 31 | \( 1 - 9.42T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 5.55T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.309132718445432569889628604625, −6.66337222756776950049802985957, −6.34918110140700006449184928586, −6.08283918526878293697565440420, −5.00696175417455066303039734763, −4.74459951631265223036152899118, −3.28790061890838817075531993900, −2.49419237571071710811154161531, −1.66442869284158027088043616601, 0,
1.66442869284158027088043616601, 2.49419237571071710811154161531, 3.28790061890838817075531993900, 4.74459951631265223036152899118, 5.00696175417455066303039734763, 6.08283918526878293697565440420, 6.34918110140700006449184928586, 6.66337222756776950049802985957, 8.309132718445432569889628604625
