L(s) = 1 | + (−0.296 + 1.70i)3-s + 5-s − 4.75i·7-s + (−2.82 − 1.01i)9-s − 4.74·11-s + 2.55i·13-s + (−0.296 + 1.70i)15-s + 2.48i·17-s + 1.87·19-s + (8.12 + 1.41i)21-s + 0.122i·23-s + 25-s + (2.56 − 4.51i)27-s + 5.57i·29-s − 1.05i·31-s + ⋯ |
L(s) = 1 | + (−0.171 + 0.985i)3-s + 0.447·5-s − 1.79i·7-s + (−0.941 − 0.337i)9-s − 1.43·11-s + 0.707i·13-s + (−0.0766 + 0.440i)15-s + 0.602i·17-s + 0.430·19-s + (1.77 + 0.308i)21-s + 0.0254i·23-s + 0.200·25-s + (0.493 − 0.869i)27-s + 1.03i·29-s − 0.190i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0233 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0233 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.234958794\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.234958794\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.296 - 1.70i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-8.02 + 1.59i)T \) |
good | 7 | \( 1 + 4.75iT - 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 2.55iT - 13T^{2} \) |
| 17 | \( 1 - 2.48iT - 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 0.122iT - 23T^{2} \) |
| 29 | \( 1 - 5.57iT - 29T^{2} \) |
| 31 | \( 1 + 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 - 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 4.59iT - 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.30iT - 59T^{2} \) |
| 61 | \( 1 + 0.253iT - 61T^{2} \) |
| 71 | \( 1 - 5.92iT - 71T^{2} \) |
| 73 | \( 1 - 9.97T + 73T^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 - 6.91iT - 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−8.698656309553667395225084001207, −7.890208305106731277325291442247, −7.16714326025884080823413012840, −6.44655041945539448544766160713, −5.41787650898816363352947631916, −4.90556014552281547330407735240, −4.02077095113097948032721854777, −3.46404708871938812010385042088, −2.33963629492047951818873330850, −0.942748526803813968231107486727,
0.42079932462504735835311593407, 1.95693923839260387004432069221, 2.54538439099515922129965726716, 3.13692043492502025401450847406, 4.93185542089255652332573084284, 5.54655696988494896872954155967, 5.79091044208297993352082764496, 6.76602828545164951613509145227, 7.61927309877785279033299713331, 8.232628587240254436835747394615