L(s) = 1 | + 1.77·3-s + 5-s + 2.66·7-s + 0.148·9-s + 2.35·11-s + 0.522·13-s + 1.77·15-s + 5.20·17-s + 5.03·19-s + 4.73·21-s + 2.63·23-s + 25-s − 5.05·27-s + 0.177·29-s − 6.99·31-s + 4.17·33-s + 2.66·35-s − 37-s + 0.927·39-s − 4.87·41-s − 12.0·43-s + 0.148·45-s − 1.98·47-s + 0.124·49-s + 9.23·51-s + 3.19·53-s + 2.35·55-s + ⋯ |
L(s) = 1 | + 1.02·3-s + 0.447·5-s + 1.00·7-s + 0.0495·9-s + 0.709·11-s + 0.144·13-s + 0.458·15-s + 1.26·17-s + 1.15·19-s + 1.03·21-s + 0.548·23-s + 0.200·25-s − 0.973·27-s + 0.0330·29-s − 1.25·31-s + 0.726·33-s + 0.451·35-s − 0.164·37-s + 0.148·39-s − 0.760·41-s − 1.83·43-s + 0.0221·45-s − 0.289·47-s + 0.0178·49-s + 1.29·51-s + 0.439·53-s + 0.317·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.554856150\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.554856150\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 1.77T + 3T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.522T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 0.177T + 29T^{2} \) |
| 31 | \( 1 + 6.99T + 31T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 1.98T + 47T^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 0.201T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 - 7.87T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.709094197169856061812893284098, −8.063281060846413612356462676443, −7.47208637373524309400670910628, −6.58844272939668861806368620119, −5.46536403956889888055530070172, −5.03175813916357883387821787601, −3.69567162456136255185510272759, −3.20352268822664926872059903887, −2.01553270786878698410248675393, −1.25354886023109796862706243233,
1.25354886023109796862706243233, 2.01553270786878698410248675393, 3.20352268822664926872059903887, 3.69567162456136255185510272759, 5.03175813916357883387821787601, 5.46536403956889888055530070172, 6.58844272939668861806368620119, 7.47208637373524309400670910628, 8.063281060846413612356462676443, 8.709094197169856061812893284098