L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.500 + 0.866i)8-s + (0.0603 − 0.342i)11-s + (0.173 − 0.984i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (0.173 + 0.984i)32-s + (1.17 + 0.984i)34-s + (−0.326 + 1.85i)38-s + (1.76 + 0.642i)41-s + (0.266 − 1.50i)43-s + (−0.173 − 0.300i)44-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.500 + 0.866i)8-s + (0.0603 − 0.342i)11-s + (0.173 − 0.984i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (0.173 + 0.984i)32-s + (1.17 + 0.984i)34-s + (−0.326 + 1.85i)38-s + (1.76 + 0.642i)41-s + (0.266 − 1.50i)43-s + (−0.173 − 0.300i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6894936804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6894936804\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 11 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 13 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
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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
−9.287754009308951798588358003772, −8.665547324002765287191858763053, −7.57938401035405872648975529956, −7.19199280039092111793815717043, −6.29000777858424247564983748595, −5.42231883886997333128942299133, −4.58815469625941458868910049291, −3.10424445265575968068013332402, −2.23806237755771189090842454127, −0.72109630646768855216018339324,
1.40741937195249149060067017829, 2.34076913402442625828128638181, 3.57093019607482095583095089548, 4.28360725707797285438794421648, 5.80551543510447662452795523648, 6.32775618109573510465973524514, 7.48960917540987274172568766491, 7.900166228812693776623074321368, 8.763905773921337214143773541337, 9.491324549225981955366593136932