| L(s) = 1 | − 4·5-s + 2·7-s + 11-s + 17-s − 6·23-s + 11·25-s + 2·29-s − 4·31-s − 8·35-s + 2·37-s + 6·41-s + 4·43-s − 6·47-s − 3·49-s − 8·53-s − 4·55-s + 8·59-s − 8·61-s + 4·67-s − 6·71-s + 10·73-s + 2·77-s + 6·79-s − 4·83-s − 4·85-s + 14·89-s + 14·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 0.301·11-s + 0.242·17-s − 1.25·23-s + 11/5·25-s + 0.371·29-s − 0.718·31-s − 1.35·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.875·47-s − 3/7·49-s − 1.09·53-s − 0.539·55-s + 1.04·59-s − 1.02·61-s + 0.488·67-s − 0.712·71-s + 1.17·73-s + 0.227·77-s + 0.675·79-s − 0.439·83-s − 0.433·85-s + 1.48·89-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.67537027226808, −14.88525794230463, −14.58253656956762, −14.20412977051116, −13.38022735728802, −12.67151567832106, −12.19235459933128, −11.83067058787357, −11.17421608637236, −10.99513404121877, −10.22346191785817, −9.469711444601915, −8.861354970072107, −8.172877953881889, −7.850363581038918, −7.487517332040004, −6.684844009300580, −6.087744886563086, −5.165037718673793, −4.638093520336587, −4.010235170571225, −3.605509835615766, −2.783541391415216, −1.827642251366904, −0.9153336593783038, 0,
0.9153336593783038, 1.827642251366904, 2.783541391415216, 3.605509835615766, 4.010235170571225, 4.638093520336587, 5.165037718673793, 6.087744886563086, 6.684844009300580, 7.487517332040004, 7.850363581038918, 8.172877953881889, 8.861354970072107, 9.469711444601915, 10.22346191785817, 10.99513404121877, 11.17421608637236, 11.83067058787357, 12.19235459933128, 12.67151567832106, 13.38022735728802, 14.20412977051116, 14.58253656956762, 14.88525794230463, 15.67537027226808
