| L(s) = 1 | + (1.41 + i)3-s + 2.82·5-s − 2.82i·7-s + (1.00 + 2.82i)9-s − 2i·11-s − 4i·13-s + (4.00 + 2.82i)15-s + 5.65i·17-s − 2.82·19-s + (2.82 − 4.00i)21-s − 8·23-s + 3.00·25-s + (−1.41 + 5.00i)27-s + 2.82·29-s + 8.48i·31-s + ⋯ |
| L(s) = 1 | + (0.816 + 0.577i)3-s + 1.26·5-s − 1.06i·7-s + (0.333 + 0.942i)9-s − 0.603i·11-s − 1.10i·13-s + (1.03 + 0.730i)15-s + 1.37i·17-s − 0.648·19-s + (0.617 − 0.872i)21-s − 1.66·23-s + 0.600·25-s + (−0.272 + 0.962i)27-s + 0.525·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05664 + 0.175152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05664 + 0.175152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.79914849491868494790316838445, −10.35128972881533121549014464069, −9.780836257400744397602104736869, −8.532683396691953250098074008156, −7.947766604517057122997277895903, −6.51420947557279667385486354534, −5.55266856522322212854624847045, −4.25126178719914202565305768732, −3.17912207350930069820924720373, −1.73879449695613555507946971694,
1.97827223360558049360806343969, 2.49080542961897640025428532955, 4.31013749661651041595736814992, 5.75888505131490339091231454492, 6.49874924579374396926734622456, 7.58377505561171798852967983053, 8.762863510823174193409748027467, 9.429259669666612344743670389477, 9.957439583505987041020275147524, 11.59476806842630171091770021624
