L(s) = 1 | + (0.618 − 1.61i)3-s + 1.23·5-s + 3.23i·7-s + (−2.23 − 2.00i)9-s − 0.763i·11-s − 4.47i·13-s + (0.763 − 2.00i)15-s − 6.47i·17-s + 5.23·19-s + (5.23 + 2.00i)21-s + 6.47·23-s − 3.47·25-s + (−4.61 + 2.38i)27-s + 9.23·29-s + 0.763i·31-s + ⋯ |
L(s) = 1 | + (0.356 − 0.934i)3-s + 0.552·5-s + 1.22i·7-s + (−0.745 − 0.666i)9-s − 0.230i·11-s − 1.24i·13-s + (0.197 − 0.516i)15-s − 1.56i·17-s + 1.20·19-s + (1.14 + 0.436i)21-s + 1.34·23-s − 0.694·25-s + (−0.888 + 0.458i)27-s + 1.71·29-s + 0.137i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56985 - 1.01762i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56985 - 1.01762i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 0.763iT - 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 - 0.763iT - 31T^{2} \) |
| 37 | \( 1 + 0.472iT - 37T^{2} \) |
| 41 | \( 1 + 2.47iT - 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 3.23iT - 59T^{2} \) |
| 61 | \( 1 - 8.47iT - 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 13.7iT - 79T^{2} \) |
| 83 | \( 1 - 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−9.933578679137095816712990570155, −9.203463525811448549476932953378, −8.483212531289877669703548751814, −7.59798614506119529994196141514, −6.71368811642471717302431040962, −5.66507361989607478536249095400, −5.17121339150905459758234223314, −3.03795318017681575240638062702, −2.62124695530669829008843234732, −1.02081493097523189981406554888,
1.57529871934387436279978745277, 3.11232730619813553288894916702, 4.13399543037172595332335176138, 4.82030855904975558518792360855, 6.05194532522764385833944496954, 7.01588506882836258599405623548, 8.008908528838254671261920918595, 8.958271565234225082256350097080, 9.750953125497315800468477289560, 10.32318757223453145073239624383