L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.351548020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.351548020\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (1.53 + 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−8.693969863280707446424814144264, −8.404576939099660407216881528732, −7.54119766300800202011269890721, −6.85804750724784958760856829780, −5.69589936947344784667275560320, −4.72635166502805319303727712921, −3.83063502472055764933929353401, −3.09160034479100497144954793305, −1.96394389029809764817339897498, −1.51130941652604428453138521343,
0.923645274946067624005142048956, 2.82641898652551433707878187510, 3.50341200695774901342190382091, 4.50151199209770002992900826768, 5.11028872863095017786895798927, 6.15434123755423510434533787128, 6.76274923356974620945095950224, 7.64878694315030921847091771969, 8.414299334865891283984789439087, 8.915282386766325044860552216063