L(s) = 1 | − 0.465·5-s + (−0.656 − 2.56i)7-s − 5.28i·11-s − 4.50i·13-s + 3.15·17-s + 1.42i·19-s − 2.27i·23-s − 4.78·25-s + 6.09i·29-s − 2.76i·31-s + (0.305 + 1.19i)35-s − 0.613·37-s − 6.93·41-s − 3.08·43-s + 9.30·47-s + ⋯ |
L(s) = 1 | − 0.208·5-s + (−0.248 − 0.968i)7-s − 1.59i·11-s − 1.24i·13-s + 0.766·17-s + 0.327i·19-s − 0.474i·23-s − 0.956·25-s + 1.13i·29-s − 0.497i·31-s + (0.0516 + 0.201i)35-s − 0.100·37-s − 1.08·41-s − 0.470·43-s + 1.35·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9405187825\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9405187825\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.656 + 2.56i)T \) |
good | 5 | \( 1 + 0.465T + 5T^{2} \) |
| 11 | \( 1 + 5.28iT - 11T^{2} \) |
| 13 | \( 1 + 4.50iT - 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 1.42iT - 19T^{2} \) |
| 23 | \( 1 + 2.27iT - 23T^{2} \) |
| 29 | \( 1 - 6.09iT - 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 0.613T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 - 3.62T + 59T^{2} \) |
| 61 | \( 1 + 2.27iT - 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.38iT - 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−8.254698228020811576564859632338, −7.72865967116827746152631434465, −6.96728099623662843573716570879, −5.92363557495909475701715877215, −5.53319830501503744229042373969, −4.32525896951717468155003363136, −3.45579647865149687406232936070, −2.97392359401181303024623796546, −1.27465244600209958627632312032, −0.30408497865373321582086250110,
1.70202169322486778217128795937, 2.37138364825019027054632663739, 3.57247200912144244129256134431, 4.43189264998195892065874868810, 5.19235936843139461943206238683, 6.05561224078160979849189041057, 6.88993417370929853746762963106, 7.47847304850616191128572571153, 8.365670001315795373293301015644, 9.150800708795795794391766034194