L(s) = 1 | + (−1.58 + 0.707i)3-s + 1.41i·5-s + (−1 + 2.44i)7-s + (2.00 − 2.23i)9-s − 3.46·11-s − 3.16·13-s + (−1.00 − 2.23i)15-s + 3.16·19-s + (−0.150 − 4.58i)21-s − 4.47i·23-s + 2.99·25-s + (−1.58 + 4.94i)27-s − 6.92·29-s − 4.89i·31-s + (5.47 − 2.44i)33-s + ⋯ |
L(s) = 1 | + (−0.912 + 0.408i)3-s + 0.632i·5-s + (−0.377 + 0.925i)7-s + (0.666 − 0.745i)9-s − 1.04·11-s − 0.877·13-s + (−0.258 − 0.577i)15-s + 0.725·19-s + (−0.0329 − 0.999i)21-s − 0.932i·23-s + 0.599·25-s + (−0.304 + 0.952i)27-s − 1.28·29-s − 0.879i·31-s + (0.953 − 0.426i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0314707 - 0.0872976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0314707 - 0.0872976i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 8.94iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−11.05703085694122744483840564856, −10.10456598901781133498690627556, −9.666919313481853431842986579794, −8.495417863384732089943266797824, −7.31287884081940843862478824320, −6.54346014794240830411103790902, −5.50893650984159386826917870426, −4.94535813837991991589472579616, −3.45620235271945572286410989286, −2.35205801354651184525454605043,
0.05436564044930796610488716938, 1.53384758518308385213584364139, 3.27706998711262453755857357083, 4.79455491760558373971531106621, 5.21956915940024198201841609451, 6.43906596233229392640254584418, 7.40221620896111846277966712910, 7.87532313552964458583834781294, 9.313481159661557972904423815905, 10.12281959466225446137588368807