L(s) = 1 | + i·3-s + (−2 + i)5-s + i·7-s − 9-s + 4·11-s + 2i·13-s + (−1 − 2i)15-s + 6i·17-s + 6·19-s − 21-s − 2i·23-s + (3 − 4i)25-s − i·27-s + 6·29-s − 2·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.894 + 0.447i)5-s + 0.377i·7-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + (−0.258 − 0.516i)15-s + 1.45i·17-s + 1.37·19-s − 0.218·21-s − 0.417i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.613148800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613148800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.090278864648666465341224446487, −8.064299758356909863675232984616, −7.51226570486987670801716930340, −6.45812107844451681419746120773, −6.04780182219825164866892677847, −4.84014988807504099471872870284, −4.09016668267637507937135721745, −3.55108049990210595287526687804, −2.56027479478351291638804228502, −1.17505176232683934023706049905,
0.61356664369414126430990298668, 1.33082579950994232297089636410, 2.89422433923791605270181446126, 3.55966027357382780450031188523, 4.56893854666655810046859401015, 5.23231614210226916337449333408, 6.26122938090300512744682722937, 7.10805706004313871067638185457, 7.55288325098702594165502106098, 8.218989491862241874310563928510