| L(s) = 1 | + 0.517i·2-s + (−0.793 + 0.608i)3-s + 0.732·4-s + (−0.315 − 0.410i)6-s + 0.896i·8-s + (0.258 − 0.965i)9-s + (−0.580 + 0.445i)12-s − 1.98i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.545 − 0.711i)24-s + 25-s + 1.02·26-s + (0.382 + 0.923i)27-s + ⋯ |
| L(s) = 1 | + 0.517i·2-s + (−0.793 + 0.608i)3-s + 0.732·4-s + (−0.315 − 0.410i)6-s + 0.896i·8-s + (0.258 − 0.965i)9-s + (−0.580 + 0.445i)12-s − 1.98i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.545 − 0.711i)24-s + 25-s + 1.02·26-s + (0.382 + 0.923i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.238573477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238573477\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.793 - 0.608i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 2 | \( 1 - 0.517iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.98iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 - 0.261iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.98T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.21T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 - 1.58iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.860396769998343986938230750984, −7.83973819209239134987103407160, −7.49355570991053205780930767137, −6.46015278699706677221932419894, −5.79768097644276174132267189316, −5.42320536545003180626230081539, −4.45876478020779742585787556170, −3.36788821901708467559230268002, −2.58175462689674495503495203883, −0.983545210959912465355719416376,
1.15567659767427194603405097640, 1.99939003467269022957465014669, 2.83092863563031625630096300232, 4.11255412433706860817419424344, 4.79397773979071469823212204011, 5.91303124911746665399776171357, 6.56480527980357855281634446480, 7.02810326189943296709900235662, 7.67909566860291837579547313062, 8.844204963052047219907294509189
