L(s) = 1 | + 3i·3-s + 12.5·5-s + (18.2 + 2.88i)7-s − 9·9-s + 55.4·11-s + 92.1·13-s + 37.5i·15-s + 118. i·17-s + 155. i·19-s + (−8.66 + 54.8i)21-s − 125. i·23-s + 31.6·25-s − 27i·27-s + 131. i·29-s − 66.0·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.11·5-s + (0.987 + 0.155i)7-s − 0.333·9-s + 1.51·11-s + 1.96·13-s + 0.646i·15-s + 1.68i·17-s + 1.87i·19-s + (−0.0900 + 0.570i)21-s − 1.13i·23-s + 0.253·25-s − 0.192i·27-s + 0.842i·29-s − 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.881688195\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.881688195\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (-18.2 - 2.88i)T \) |
good | 5 | \( 1 - 12.5T + 125T^{2} \) |
| 11 | \( 1 - 55.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 125. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 66.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 20.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 355.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 79.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 580. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 496.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 232. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 551. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 437. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 191. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 93.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 758. iT - 9.12e5T^{2} \) |
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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
−9.350759261943019966450088895307, −8.482938885727058851038782358576, −8.279516896585644390813965598056, −6.52109720132618743130889278174, −6.09140567393966279587335938932, −5.36263569958889021368168257620, −4.03602542661140418654681815044, −3.63535765947389657674283400020, −1.71014046658485481046886993298, −1.50100015198367664218127040220,
0.993865546873576556899756578305, 1.51075988532183141874963265052, 2.67522256887267913096609204199, 3.92245205113131901937446173575, 5.01904455863587628658554802998, 5.85939757803443469670503887843, 6.64229657008665734265944656749, 7.28766923212592547665006921634, 8.448642194719918332120695996889, 9.091527620121302848341469018100