L(s) = 1 | + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (4.95 + 0.632i)5-s − 2.44·6-s + (3.11 + 3.11i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (4.32 + 5.59i)10-s − 11.9·11-s + (−2.44 − 2.44i)12-s + (−6.79 + 6.79i)13-s + 6.22i·14-s + (−6.84 + 5.29i)15-s − 4·16-s + (−3.44 − 3.44i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.991 + 0.126i)5-s − 0.408·6-s + (0.444 + 0.444i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.432 + 0.559i)10-s − 1.08·11-s + (−0.204 − 0.204i)12-s + (−0.522 + 0.522i)13-s + 0.444i·14-s + (−0.456 + 0.353i)15-s − 0.250·16-s + (−0.202 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.887178303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887178303\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (-4.95 - 0.632i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (-3.11 - 3.11i)T + 49iT^{2} \) |
| 11 | \( 1 + 11.9T + 121T^{2} \) |
| 13 | \( 1 + (6.79 - 6.79i)T - 169iT^{2} \) |
| 17 | \( 1 + (3.44 + 3.44i)T + 289iT^{2} \) |
| 19 | \( 1 - 27.8iT - 361T^{2} \) |
| 29 | \( 1 - 26.4iT - 841T^{2} \) |
| 31 | \( 1 + 16.0T + 961T^{2} \) |
| 37 | \( 1 + (-13.6 - 13.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (22.4 - 22.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (35.7 + 35.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (41.6 - 41.6i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 59.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 9.79T + 3.72e3T^{2} \) |
| 67 | \( 1 + (35.1 + 35.1i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 28.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (2.27 - 2.27i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 21.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (48.0 - 48.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-43.1 - 43.1i)T + 9.40e3iT^{2} \) |
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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
−10.57993264417772615877212466428, −9.854556586317704185743666969037, −8.941533323940449931825489626632, −7.971142669046654403943862794178, −6.94099977444772066650933115047, −5.94350324773634849644070502970, −5.32009768388453833114410840148, −4.56189528649878761103909809939, −3.09412619986553642586811227689, −1.89838269320168605418812694933,
0.56275482508573433535854153166, 1.98344996791271474542022324897, 2.88112962451590983807646488597, 4.60136657470196094816020746131, 5.22247496466544946554122466285, 6.09533353344239766840789889818, 7.12494989186784432639245085815, 8.047509645655380199632950011469, 9.275844814788796041207761679217, 10.10173555663596261504255869129