L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.69 + 0.336i)3-s + 1.00i·4-s + (2.14 + 0.630i)5-s + (1.43 + 0.963i)6-s + (0.804 − 0.804i)7-s + (0.707 − 0.707i)8-s + (2.77 − 1.14i)9-s + (−1.07 − 1.96i)10-s + 3.15i·11-s + (−0.336 − 1.69i)12-s + (−2.72 − 2.72i)13-s − 1.13·14-s + (−3.85 − 0.348i)15-s − 1.00·16-s + (2.04 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.980 + 0.194i)3-s + 0.500i·4-s + (0.959 + 0.282i)5-s + (0.587 + 0.393i)6-s + (0.304 − 0.304i)7-s + (0.250 − 0.250i)8-s + (0.924 − 0.381i)9-s + (−0.338 − 0.620i)10-s + 0.951i·11-s + (−0.0972 − 0.490i)12-s + (−0.757 − 0.757i)13-s − 0.304·14-s + (−0.995 − 0.0900i)15-s − 0.250·16-s + (0.495 + 0.495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.990025 + 0.0702257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.990025 + 0.0702257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.69 - 0.336i)T \) |
| 5 | \( 1 + (-2.14 - 0.630i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-0.804 + 0.804i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.15iT - 11T^{2} \) |
| 13 | \( 1 + (2.72 + 2.72i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.04 - 2.04i)T + 17iT^{2} \) |
| 23 | \( 1 + (1.82 - 1.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + (-7.25 + 7.25i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.31iT - 41T^{2} \) |
| 43 | \( 1 + (-0.643 - 0.643i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.22 - 3.22i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 + (-3.86 + 3.86i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.944iT - 71T^{2} \) |
| 73 | \( 1 + (-3.61 - 3.61i)T + 73iT^{2} \) |
| 79 | \( 1 - 15.0iT - 79T^{2} \) |
| 83 | \( 1 + (5.54 - 5.54i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.27T + 89T^{2} \) |
| 97 | \( 1 + (-2.89 + 2.89i)T - 97iT^{2} \) |
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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
−10.58972282092446129306235174735, −9.948594660301691658072773659492, −9.574531397719601661048612681067, −8.044929043403174299310061654930, −7.17793117249252486374070327337, −6.17952511929787469415641939698, −5.20172599230014568611463436310, −4.19627625483014069967655722972, −2.57748772419066561468882708136, −1.22202518612799444503752379923,
0.927319047412614867296605421999, 2.37644739662317356109573184196, 4.59718070963898811728742475904, 5.37364439624587312231337135660, 6.19890036147475100618174196000, 6.90048063611678536310173404184, 8.070128824851386469813958079616, 8.992566421187973194899589123729, 9.908102546130311440793725932012, 10.50596618465942416842953552953