| L(s) = 1 | + 3-s − 0.858·5-s + 2.33·7-s + 9-s + 0.664·11-s − 2.80·13-s − 0.858·15-s − 4.73·17-s − 7.58·19-s + 2.33·21-s + 6.58·23-s − 4.26·25-s + 27-s − 4.75·29-s − 2.96·31-s + 0.664·33-s − 2.00·35-s + 4.12·37-s − 2.80·39-s − 7.76·41-s + 4.13·43-s − 0.858·45-s − 1.39·47-s − 1.56·49-s − 4.73·51-s − 1.74·53-s − 0.569·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.383·5-s + 0.881·7-s + 0.333·9-s + 0.200·11-s − 0.777·13-s − 0.221·15-s − 1.14·17-s − 1.74·19-s + 0.508·21-s + 1.37·23-s − 0.852·25-s + 0.192·27-s − 0.883·29-s − 0.531·31-s + 0.115·33-s − 0.338·35-s + 0.678·37-s − 0.449·39-s − 1.21·41-s + 0.631·43-s − 0.127·45-s − 0.202·47-s − 0.222·49-s − 0.663·51-s − 0.239·53-s − 0.0768·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.858T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 - 0.664T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 + 1.39T + 47T^{2} \) |
| 53 | \( 1 + 1.74T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.762T + 61T^{2} \) |
| 67 | \( 1 - 0.00130T + 67T^{2} \) |
| 71 | \( 1 + 2.99T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.082366029677731649925088905707, −7.45598752554869832252201879124, −6.79663946881233899451856346124, −5.89574581005216367969722322807, −4.71599602193156833328120926482, −4.44322900350144354987282229121, −3.42954198585822308054046392617, −2.35119402809370291308673280722, −1.68017941238334238094147569791, 0,
1.68017941238334238094147569791, 2.35119402809370291308673280722, 3.42954198585822308054046392617, 4.44322900350144354987282229121, 4.71599602193156833328120926482, 5.89574581005216367969722322807, 6.79663946881233899451856346124, 7.45598752554869832252201879124, 8.082366029677731649925088905707
