| L(s) = 1 | − 1.38·2-s − 0.0848·4-s − 4.17·5-s + 3.01·7-s + 2.88·8-s + 5.78·10-s + 11-s + 0.616·13-s − 4.17·14-s − 3.82·16-s − 3.75·17-s + 5.19·19-s + 0.354·20-s − 1.38·22-s − 1.07·23-s + 12.4·25-s − 0.853·26-s − 0.256·28-s − 4.93·29-s − 5.32·31-s − 0.479·32-s + 5.20·34-s − 12.6·35-s − 3.81·37-s − 7.19·38-s − 12.0·40-s + 2.12·41-s + ⋯ |
| L(s) = 1 | − 0.978·2-s − 0.0424·4-s − 1.86·5-s + 1.14·7-s + 1.02·8-s + 1.82·10-s + 0.301·11-s + 0.170·13-s − 1.11·14-s − 0.955·16-s − 0.911·17-s + 1.19·19-s + 0.0792·20-s − 0.295·22-s − 0.223·23-s + 2.49·25-s − 0.167·26-s − 0.0484·28-s − 0.915·29-s − 0.956·31-s − 0.0847·32-s + 0.891·34-s − 2.13·35-s − 0.626·37-s − 1.16·38-s − 1.90·40-s + 0.331·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 13 | \( 1 - 0.616T + 13T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 - 4.50T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 9.81T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.943605258978542477842863347543, −7.33496365132506389938057687876, −6.78120668705070415975772121517, −5.31511446346003991534326697292, −4.74957479377346858568863809352, −4.01691652708691451053058799339, −3.45175246735068129611805444913, −1.98195835147847701749412079733, −1.01025495061394091344880587581, 0,
1.01025495061394091344880587581, 1.98195835147847701749412079733, 3.45175246735068129611805444913, 4.01691652708691451053058799339, 4.74957479377346858568863809352, 5.31511446346003991534326697292, 6.78120668705070415975772121517, 7.33496365132506389938057687876, 7.943605258978542477842863347543
