L(s) = 1 | + (−1 − 2i)5-s − i·7-s + 2·11-s + 6i·13-s + 4i·17-s − 6·19-s − 8i·23-s + (−3 + 4i)25-s + 6·29-s + 2·31-s + (−2 + i)35-s − 4i·37-s − 2·41-s − 4i·43-s − 8i·47-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)5-s − 0.377i·7-s + 0.603·11-s + 1.66i·13-s + 0.970i·17-s − 1.37·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s + 1.11·29-s + 0.359·31-s + (−0.338 + 0.169i)35-s − 0.657i·37-s − 0.312·41-s − 0.609i·43-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7040098125\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7040098125\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.207059025462696215488154000710, −7.02850262169755123748291515589, −6.61618962127318062346026291891, −5.85916611282324796695007356060, −4.55586544352857365544577275102, −4.40102392997775245018299686935, −3.67504785331634815609148757602, −2.24609220186170019907254664753, −1.45628896097583109959931681246, −0.19784682898305887768992090193,
1.22128118614708430747711197718, 2.64800743778501957148776515894, 3.05759610798188764377708571982, 3.99615050969032645169286636816, 4.86791495551848823663244716611, 5.77624094091176599376352957521, 6.37978254516918660030836575865, 7.13718687999467398681961227892, 7.84557151625693392196762749095, 8.364793209251106933293610444318