L(s) = 1 | − 0.665i·5-s + i·7-s + 2.07·11-s − 5.55·13-s − 2.16i·17-s + 4.49i·19-s − 4.28·23-s + 4.55·25-s − 1.41i·29-s − 6.61i·31-s + 0.665·35-s + 5.43·37-s − 5.69i·41-s − 2.11i·43-s + 10.5·47-s + ⋯ |
L(s) = 1 | − 0.297i·5-s + 0.377i·7-s + 0.627·11-s − 1.54·13-s − 0.524i·17-s + 1.03i·19-s − 0.892·23-s + 0.911·25-s − 0.262i·29-s − 1.18i·31-s + 0.112·35-s + 0.894·37-s − 0.889i·41-s − 0.322i·43-s + 1.53·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.636533414\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636533414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.665iT - 5T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 + 2.16iT - 17T^{2} \) |
| 19 | \( 1 - 4.49iT - 19T^{2} \) |
| 23 | \( 1 + 4.28T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 6.61iT - 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 5.69iT - 41T^{2} \) |
| 43 | \( 1 + 2.11iT - 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 + 14.0iT - 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.824iT - 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 - 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 0.824T + 97T^{2} \) |
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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
−8.360060573773636413830536576807, −7.61235166304192396688881058173, −7.02722927791593856247181580061, −6.03433489787495981011384294550, −5.46341489539800674243376720020, −4.54590480520091139118770791005, −3.90328083744405516378406358068, −2.70455296958549551120806528462, −1.98933741999365350588236977846, −0.60285681886471432230095864836,
0.875278314986834915823927579903, 2.17784153476863363437713306591, 2.97189243091398691759292514096, 3.99428340275457118491062983248, 4.71472347883789054495907779576, 5.46924620279390791921142056301, 6.56150049596875855719387815083, 6.96575300745726664277794480518, 7.67811400812115790353488572052, 8.547200039390650689585584996179