L(s) = 1 | − 2.08i·2-s − i·3-s − 2.34·4-s + 0.833i·5-s − 2.08·6-s + (−2.19 − 1.47i)7-s + 0.726i·8-s − 9-s + 1.73·10-s + (−1.23 − 3.07i)11-s + 2.34i·12-s − 4.54·13-s + (−3.06 + 4.58i)14-s + 0.833·15-s − 3.18·16-s + 6.38·17-s + ⋯ |
L(s) = 1 | − 1.47i·2-s − 0.577i·3-s − 1.17·4-s + 0.372i·5-s − 0.851·6-s + (−0.830 − 0.556i)7-s + 0.256i·8-s − 0.333·9-s + 0.549·10-s + (−0.372 − 0.927i)11-s + 0.677i·12-s − 1.26·13-s + (−0.820 + 1.22i)14-s + 0.215·15-s − 0.795·16-s + 1.54·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.100835 + 0.965716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100835 + 0.965716i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.19 + 1.47i)T \) |
| 11 | \( 1 + (1.23 + 3.07i)T \) |
good | 2 | \( 1 + 2.08iT - 2T^{2} \) |
| 5 | \( 1 - 0.833iT - 5T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 9.16T + 23T^{2} \) |
| 29 | \( 1 - 0.152iT - 29T^{2} \) |
| 31 | \( 1 + 8.36iT - 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 - 5.66iT - 43T^{2} \) |
| 47 | \( 1 + 6.00iT - 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 + 1.05iT - 59T^{2} \) |
| 61 | \( 1 + 8.34T + 61T^{2} \) |
| 67 | \( 1 - 4.33T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.70iT - 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 8.13iT - 89T^{2} \) |
| 97 | \( 1 - 17.7iT - 97T^{2} \) |
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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
−11.71115283961367719470101084734, −10.78701926696800628837517925560, −10.02771339132157164298025577778, −9.173701904179658010868780914039, −7.64843356780059979619770464078, −6.73849888791643757734024808379, −5.20666618238337602689880857822, −3.42151740264107220777556624677, −2.76308393440092093586973449869, −0.819702507730140086831793935414,
3.01990878114180841235744627087, 5.00493812745973666519207587125, 5.28800919761663986095408379456, 6.80100004264128846138300260253, 7.51347874961460277221674194298, 8.789281776504179070333591911803, 9.490298831037440267500574707241, 10.42489092849739842633806316558, 12.08846867648581670545370772339, 12.69657736672843398999965041124