| L(s) = 1 | − 2.30·2-s + 0.302·3-s + 3.30·4-s − 3·5-s − 0.697·6-s + 7-s − 3.00·8-s − 2.90·9-s + 6.90·10-s − 0.697·11-s + 1.00·12-s − 5.60·13-s − 2.30·14-s − 0.908·15-s + 0.302·16-s − 5.30·17-s + 6.69·18-s + 19-s − 9.90·20-s + 0.302·21-s + 1.60·22-s − 3·23-s − 0.908·24-s + 4·25-s + 12.9·26-s − 1.78·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.174·3-s + 1.65·4-s − 1.34·5-s − 0.284·6-s + 0.377·7-s − 1.06·8-s − 0.969·9-s + 2.18·10-s − 0.210·11-s + 0.288·12-s − 1.55·13-s − 0.615·14-s − 0.234·15-s + 0.0756·16-s − 1.28·17-s + 1.57·18-s + 0.229·19-s − 2.21·20-s + 0.0660·21-s + 0.342·22-s − 0.625·23-s − 0.185·24-s + 0.800·25-s + 2.53·26-s − 0.344·27-s + 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 0.302T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 + 0.697T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 0.697T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−12.16223670351254947918265846916, −11.52537045404929951865303175576, −10.62247660128235055065423533368, −9.443489214838429577661138590233, −8.339758778433576146179874636956, −7.86986330637094473016029712752, −6.76314225479089571181907210672, −4.64573456595693886946460072810, −2.60154666418571372875098416271, 0,
2.60154666418571372875098416271, 4.64573456595693886946460072810, 6.76314225479089571181907210672, 7.86986330637094473016029712752, 8.339758778433576146179874636956, 9.443489214838429577661138590233, 10.62247660128235055065423533368, 11.52537045404929951865303175576, 12.16223670351254947918265846916
