L(s) = 1 | + 0.514·2-s − 3-s − 1.73·4-s − 0.514·6-s + 1.81·7-s − 1.92·8-s + 9-s + 1.06·11-s + 1.73·12-s − 1.63·13-s + 0.932·14-s + 2.47·16-s + 6.28·17-s + 0.514·18-s − 7.69·19-s − 1.81·21-s + 0.546·22-s + 6.24·23-s + 1.92·24-s − 0.843·26-s − 27-s − 3.14·28-s − 1.20·29-s + 4.02·31-s + 5.12·32-s − 1.06·33-s + 3.23·34-s + ⋯ |
L(s) = 1 | + 0.364·2-s − 0.577·3-s − 0.867·4-s − 0.210·6-s + 0.684·7-s − 0.679·8-s + 0.333·9-s + 0.320·11-s + 0.500·12-s − 0.454·13-s + 0.249·14-s + 0.619·16-s + 1.52·17-s + 0.121·18-s − 1.76·19-s − 0.395·21-s + 0.116·22-s + 1.30·23-s + 0.392·24-s − 0.165·26-s − 0.192·27-s − 0.593·28-s − 0.224·29-s + 0.723·31-s + 0.905·32-s − 0.184·33-s + 0.554·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.543885701\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.543885701\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
good | 2 | \( 1 - 0.514T + 2T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 7.38T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 0.0182T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−7.902427032047932888819656280543, −7.09117473989696794416759394441, −6.26444978133691504382312102836, −5.63857877863339058618477649699, −4.92488314335744388812617618116, −4.50326972642553439010268111776, −3.71128514774569970687111687683, −2.82305294362512303301115505162, −1.59606425910783968269463515152, −0.63364280307848728191314761780,
0.63364280307848728191314761780, 1.59606425910783968269463515152, 2.82305294362512303301115505162, 3.71128514774569970687111687683, 4.50326972642553439010268111776, 4.92488314335744388812617618116, 5.63857877863339058618477649699, 6.26444978133691504382312102836, 7.09117473989696794416759394441, 7.902427032047932888819656280543