L(s) = 1 | + (−0.541 − 1.30i)2-s + (−1.30 − 1.13i)3-s + (−1.41 + 1.41i)4-s + (−2.10 + 0.765i)5-s + (−0.778 + 2.32i)6-s − 2.27·7-s + (2.61 + 1.08i)8-s + (0.414 + 2.97i)9-s + (2.13 + 2.33i)10-s − 4.20i·11-s + (3.45 − 0.239i)12-s − 3.21·13-s + (1.23 + 2.97i)14-s + (3.61 + 1.38i)15-s − 4i·16-s − 1.53·17-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.754 − 0.656i)3-s + (−0.707 + 0.707i)4-s + (−0.939 + 0.342i)5-s + (−0.317 + 0.948i)6-s − 0.859·7-s + (0.923 + 0.382i)8-s + (0.138 + 0.990i)9-s + (0.675 + 0.737i)10-s − 1.26i·11-s + (0.997 − 0.0692i)12-s − 0.891·13-s + (0.328 + 0.794i)14-s + (0.933 + 0.358i)15-s − i·16-s − 0.371·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0561969 + 0.140802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0561969 + 0.140802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.541 + 1.30i)T \) |
| 3 | \( 1 + (1.30 + 1.13i)T \) |
| 5 | \( 1 + (2.10 - 0.765i)T \) |
good | 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 + 3.21T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 1.08iT - 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 - 2.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.70iT - 43T^{2} \) |
| 47 | \( 1 - 1.08iT - 47T^{2} \) |
| 53 | \( 1 + 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.20iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.27iT - 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 4.54iT - 73T^{2} \) |
| 79 | \( 1 + 0.485iT - 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 8.40iT - 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−12.72346763953536553181892414278, −11.60287457379997192678695401041, −11.08589337107232167185511289760, −9.975377592129621485669146997510, −8.502225004914504743072371385419, −7.49361558743166755138278941098, −6.21326606944138263951129373135, −4.36108264110275123978390467498, −2.78618473632705026028466914791, −0.19621024229217829948481741662,
4.11693125411179153206936073441, 5.03258589969553009185304166927, 6.53214444788118060689317435764, 7.41527878555189654546206572994, 8.921409501696274229198855449340, 9.795875636476771573357109276645, 10.74396157971173393110827098010, 12.19998986721175847357127509675, 12.88480415649350000331614306675, 14.66442085357218251923875634171