L(s) = 1 | − 1.21·2-s − 3-s − 0.522·4-s + 1.83·5-s + 1.21·6-s − 2.18·7-s + 3.06·8-s + 9-s − 2.22·10-s − 4.11·11-s + 0.522·12-s + 2.89·13-s + 2.65·14-s − 1.83·15-s − 2.68·16-s − 2.75·17-s − 1.21·18-s + 4.55·19-s − 0.955·20-s + 2.18·21-s + 4.99·22-s − 8.48·23-s − 3.06·24-s − 1.64·25-s − 3.52·26-s − 27-s + 1.14·28-s + ⋯ |
L(s) = 1 | − 0.859·2-s − 0.577·3-s − 0.261·4-s + 0.818·5-s + 0.496·6-s − 0.825·7-s + 1.08·8-s + 0.333·9-s − 0.703·10-s − 1.23·11-s + 0.150·12-s + 0.803·13-s + 0.709·14-s − 0.472·15-s − 0.670·16-s − 0.668·17-s − 0.286·18-s + 1.04·19-s − 0.213·20-s + 0.476·21-s + 1.06·22-s − 1.76·23-s − 0.625·24-s − 0.329·25-s − 0.691·26-s − 0.192·27-s + 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 3.59T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−10.57596627597327743246876334291, −9.907351094250031471994850267526, −9.328207685083987351826704743375, −8.196812200171199023402863754620, −7.24279852719451662190165713355, −6.00365731393032492292732215274, −5.27245644487825874142831058218, −3.77354264316338176067292941958, −1.91729121926422084500442340950, 0,
1.91729121926422084500442340950, 3.77354264316338176067292941958, 5.27245644487825874142831058218, 6.00365731393032492292732215274, 7.24279852719451662190165713355, 8.196812200171199023402863754620, 9.328207685083987351826704743375, 9.907351094250031471994850267526, 10.57596627597327743246876334291