| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s + 4·13-s − 2·14-s + 16-s + 6·17-s − 18-s − 4·19-s − 2·21-s − 22-s − 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s + 2·28-s − 6·29-s − 32-s − 33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.63049693876854, −14.11866631939155, −13.38545417182462, −12.98323841021487, −12.30834002422398, −11.78714802872421, −11.41665779346423, −11.02681838100025, −10.43408433575564, −9.921630425386471, −9.500302309807247, −8.846028173346778, −8.181937000261413, −7.829101205682158, −7.503116806946150, −6.525302254082838, −6.082891502893094, −5.795491597278805, −5.039392460860705, −4.260843669064462, −3.800674318849691, −3.105416592374674, −2.018922290816023, −1.639242748693367, −0.9302797465223217, 0,
0.9302797465223217, 1.639242748693367, 2.018922290816023, 3.105416592374674, 3.800674318849691, 4.260843669064462, 5.039392460860705, 5.795491597278805, 6.082891502893094, 6.525302254082838, 7.503116806946150, 7.829101205682158, 8.181937000261413, 8.846028173346778, 9.500302309807247, 9.921630425386471, 10.43408433575564, 11.02681838100025, 11.41665779346423, 11.78714802872421, 12.30834002422398, 12.98323841021487, 13.38545417182462, 14.11866631939155, 14.63049693876854
