| L(s) = 1 | + 0.456·2-s − 1.79·4-s + 5-s − 4.37·7-s − 1.73·8-s + 0.456·10-s − 0.913·13-s − 1.99·14-s + 2.79·16-s + 1.73·17-s + 3.46·19-s − 1.79·20-s + 0.582·23-s + 25-s − 0.417·26-s + 7.84·28-s + 9.66·29-s − 8.58·31-s + 4.73·32-s + 0.791·34-s − 4.37·35-s + 7.58·37-s + 1.58·38-s − 1.73·40-s + 3.46·41-s + 9.66·43-s + 0.266·46-s + ⋯ |
| L(s) = 1 | + 0.323·2-s − 0.895·4-s + 0.447·5-s − 1.65·7-s − 0.612·8-s + 0.144·10-s − 0.253·13-s − 0.534·14-s + 0.697·16-s + 0.420·17-s + 0.794·19-s − 0.400·20-s + 0.121·23-s + 0.200·25-s − 0.0818·26-s + 1.48·28-s + 1.79·29-s − 1.54·31-s + 0.837·32-s + 0.135·34-s − 0.739·35-s + 1.24·37-s + 0.256·38-s − 0.273·40-s + 0.541·41-s + 1.47·43-s + 0.0392·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 13 | \( 1 + 0.913T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 0.582T + 23T^{2} \) |
| 29 | \( 1 - 9.66T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 9.66T + 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 + 0.818T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 4.41T + 97T^{2} \) |
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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
−7.75448580751075856928395803705, −7.03652833996812934370584313720, −6.05917868455455589890376678266, −5.84991854549980199846658892390, −4.84853783878581449964537635819, −4.12211849485841805392479364446, −3.15742600485573532902268378871, −2.80537086846802370294437404748, −1.15471173239259905997156054136, 0,
1.15471173239259905997156054136, 2.80537086846802370294437404748, 3.15742600485573532902268378871, 4.12211849485841805392479364446, 4.84853783878581449964537635819, 5.84991854549980199846658892390, 6.05917868455455589890376678266, 7.03652833996812934370584313720, 7.75448580751075856928395803705
