L(s) = 1 | + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−1.70 − 0.707i)29-s + (−0.292 − 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (−0.707 + 0.292i)53-s + 1.00·63-s + (0.707 + 0.292i)67-s + (0.707 − 1.70i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−1.70 − 0.707i)29-s + (−0.292 − 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (−0.707 + 0.292i)53-s + 1.00·63-s + (0.707 + 0.292i)67-s + (0.707 − 1.70i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.622381044\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.622381044\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.796044459468968114966148288013, −7.74119283722102375461676324619, −7.40544893255494694252710352106, −6.58375744163703846495940197755, −5.66895450037369079482594448878, −4.87983044166724496821516152477, −3.95926916136639188772365098370, −3.52202474950726525188352021695, −1.90039386967745192039866151846, −1.24491026419306281339092878023,
1.31980013440966420702442776858, 2.05716030755713228218105476889, 3.38597114131175002917153142418, 4.19410894358792032749728721280, 4.84090532637585864881173315109, 5.82334399049606068110052591746, 6.69946146839539796290721903209, 7.05629492401487632456523253250, 8.123119352432503886818050778331, 8.855873323106419223645913347163