| L(s) = 1 | − 2-s + 3·3-s − 3·4-s − 5·5-s − 3·6-s + 7·8-s + 9·9-s + 5·10-s − 9·12-s − 15·15-s + 5·16-s + 14·17-s − 9·18-s − 22·19-s + 15·20-s − 34·23-s + 21·24-s + 25·25-s + 27·27-s + 15·30-s + 2·31-s − 33·32-s − 14·34-s − 27·36-s + 22·38-s − 35·40-s − 45·45-s + ⋯ |
| L(s) = 1 | − 1/2·2-s + 3-s − 3/4·4-s − 5-s − 1/2·6-s + 7/8·8-s + 9-s + 1/2·10-s − 3/4·12-s − 15-s + 5/16·16-s + 0.823·17-s − 1/2·18-s − 1.15·19-s + 3/4·20-s − 1.47·23-s + 7/8·24-s + 25-s + 27-s + 1/2·30-s + 2/31·31-s − 1.03·32-s − 0.411·34-s − 3/4·36-s + 0.578·38-s − 7/8·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6668540075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6668540075\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 14 T + p^{2} T^{2} \) |
| 19 | \( 1 + 22 T + p^{2} T^{2} \) |
| 23 | \( 1 + 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 14 T + p^{2} T^{2} \) |
| 53 | \( 1 - 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 98 T + p^{2} T^{2} \) |
| 83 | \( 1 + 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−19.20528024791469472082930152995, −18.30636401032706904656834259710, −16.57721631502772335132152817015, −15.13715242460185422100310569876, −13.91798630493744120640438016066, −12.44366261657194403795801140514, −10.26595526686058468090529662428, −8.724899827186229663471098390093, −7.69282703936184641334875465864, −4.08039726064136068182425159874,
4.08039726064136068182425159874, 7.69282703936184641334875465864, 8.724899827186229663471098390093, 10.26595526686058468090529662428, 12.44366261657194403795801140514, 13.91798630493744120640438016066, 15.13715242460185422100310569876, 16.57721631502772335132152817015, 18.30636401032706904656834259710, 19.20528024791469472082930152995
