L(s) = 1 | − 3-s − 5·7-s + 9-s − 4·13-s + 5·17-s − 7·19-s + 5·21-s + 9·23-s − 27-s − 2·29-s + 4·31-s − 7·37-s + 4·39-s + 7·41-s − 9·47-s + 18·49-s − 5·51-s + 2·53-s + 7·57-s − 7·59-s − 2·61-s − 5·63-s + 2·67-s − 9·69-s + 5·71-s + 2·73-s + 3·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.10·13-s + 1.21·17-s − 1.60·19-s + 1.09·21-s + 1.87·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 1.15·37-s + 0.640·39-s + 1.09·41-s − 1.31·47-s + 18/7·49-s − 0.700·51-s + 0.274·53-s + 0.927·57-s − 0.911·59-s − 0.256·61-s − 0.629·63-s + 0.244·67-s − 1.08·69-s + 0.593·71-s + 0.234·73-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5701949942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5701949942\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.81959347670967, −14.65564079598913, −13.61842559124154, −13.26636496920848, −12.62635137540354, −12.40236514642426, −12.01983923770064, −11.06140526144035, −10.64104032505841, −10.14204408795865, −9.542444806268423, −9.298840309564337, −8.536201605691373, −7.734631040059185, −7.158422860881226, −6.532710445338845, −6.392185855209242, −5.470644121142104, −5.070737672735356, −4.250366843208826, −3.561698266346972, −2.943746591404192, −2.392307558709319, −1.234231706180016, −0.3087787942160484,
0.3087787942160484, 1.234231706180016, 2.392307558709319, 2.943746591404192, 3.561698266346972, 4.250366843208826, 5.070737672735356, 5.470644121142104, 6.392185855209242, 6.532710445338845, 7.158422860881226, 7.734631040059185, 8.536201605691373, 9.298840309564337, 9.542444806268423, 10.14204408795865, 10.64104032505841, 11.06140526144035, 12.01983923770064, 12.40236514642426, 12.62635137540354, 13.26636496920848, 13.61842559124154, 14.65564079598913, 14.81959347670967