L(s) = 1 | − 1.55·2-s + 0.429·4-s + 1.78·5-s − 7-s + 2.44·8-s − 2.77·10-s − 0.515·11-s − 5.03·13-s + 1.55·14-s − 4.67·16-s + 0.0785·17-s + 4.96·19-s + 0.763·20-s + 0.803·22-s − 1.02·23-s − 1.83·25-s + 7.84·26-s − 0.429·28-s − 4.62·29-s + 2.09·31-s + 2.38·32-s − 0.122·34-s − 1.78·35-s + 5.01·37-s − 7.73·38-s + 4.35·40-s + 2.22·41-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 0.214·4-s + 0.796·5-s − 0.377·7-s + 0.865·8-s − 0.877·10-s − 0.155·11-s − 1.39·13-s + 0.416·14-s − 1.16·16-s + 0.0190·17-s + 1.13·19-s + 0.170·20-s + 0.171·22-s − 0.212·23-s − 0.366·25-s + 1.53·26-s − 0.0810·28-s − 0.858·29-s + 0.376·31-s + 0.422·32-s − 0.0210·34-s − 0.300·35-s + 0.824·37-s − 1.25·38-s + 0.689·40-s + 0.346·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 + 0.515T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 - 0.0785T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 1.02T + 23T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 + 7.69T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 - 6.33T + 97T^{2} \) |
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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
−7.51082523327725661096800631848, −7.14969579231394748349786233721, −6.15845959533143997215648127230, −5.46091897735520380730094915151, −4.78540529926063898543154602355, −3.91217751865859907352404594328, −2.75406094859603973548801743570, −2.09777116581472295400036952388, −1.09405676251888750178416545756, 0,
1.09405676251888750178416545756, 2.09777116581472295400036952388, 2.75406094859603973548801743570, 3.91217751865859907352404594328, 4.78540529926063898543154602355, 5.46091897735520380730094915151, 6.15845959533143997215648127230, 7.14969579231394748349786233721, 7.51082523327725661096800631848