| L(s) = 1 | − 1.42·2-s + 3-s + 0.0397·4-s − 3.43·5-s − 1.42·6-s + 0.0177·7-s + 2.79·8-s + 9-s + 4.90·10-s − 2.14·11-s + 0.0397·12-s − 1.57·13-s − 0.0253·14-s − 3.43·15-s − 4.07·16-s − 17-s − 1.42·18-s − 1.12·19-s − 0.136·20-s + 0.0177·21-s + 3.05·22-s + 2.11·23-s + 2.79·24-s + 6.80·25-s + 2.24·26-s + 27-s + 0.000705·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 0.0198·4-s − 1.53·5-s − 0.583·6-s + 0.00670·7-s + 0.989·8-s + 0.333·9-s + 1.55·10-s − 0.645·11-s + 0.0114·12-s − 0.436·13-s − 0.00677·14-s − 0.887·15-s − 1.01·16-s − 0.242·17-s − 0.336·18-s − 0.258·19-s − 0.0305·20-s + 0.00387·21-s + 0.651·22-s + 0.441·23-s + 0.571·24-s + 1.36·25-s + 0.440·26-s + 0.192·27-s + 0.000133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 - 0.0177T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 7.22T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 5.47T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 + 8.56T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.922340078362409807871415990235, −7.09599275600880314251543124137, −6.54657702272778722132433444090, −5.04825697944317294491599454149, −4.64847427997886466716939075365, −3.85375202746429453776183599873, −3.09176357828743042022628497174, −2.15156628014833822998620856610, −0.921769575436422684557626593663, 0,
0.921769575436422684557626593663, 2.15156628014833822998620856610, 3.09176357828743042022628497174, 3.85375202746429453776183599873, 4.64847427997886466716939075365, 5.04825697944317294491599454149, 6.54657702272778722132433444090, 7.09599275600880314251543124137, 7.922340078362409807871415990235
