| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s − 7-s + 8·10-s + 2·13-s − 2·14-s − 4·16-s + 5·17-s − 5·19-s + 8·20-s + 11·25-s + 4·26-s − 2·28-s + 29-s − 3·31-s − 8·32-s + 10·34-s − 4·35-s − 8·37-s − 10·38-s − 10·41-s + 4·43-s + 8·47-s + 49-s + 22·50-s + 4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 0.377·7-s + 2.52·10-s + 0.554·13-s − 0.534·14-s − 16-s + 1.21·17-s − 1.14·19-s + 1.78·20-s + 11/5·25-s + 0.784·26-s − 0.377·28-s + 0.185·29-s − 0.538·31-s − 1.41·32-s + 1.71·34-s − 0.676·35-s − 1.31·37-s − 1.62·38-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 3.11·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.577519792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.577519792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.12202350367116, −12.67659622812228, −12.23478773902087, −11.85090623465159, −11.13360929741135, −10.56254586829116, −10.27813079963287, −9.825307108871997, −9.219221499833885, −8.769866460272373, −8.463148055302222, −7.506056856517067, −6.923235282575692, −6.484380418124123, −6.118599501188024, −5.667138817245209, −5.215856304414257, −4.960487577270820, −4.044106808247074, −3.671600659306937, −3.103699567442922, −2.496253914626628, −2.028790149510359, −1.474393205194324, −0.5974434062411748,
0.5974434062411748, 1.474393205194324, 2.028790149510359, 2.496253914626628, 3.103699567442922, 3.671600659306937, 4.044106808247074, 4.960487577270820, 5.215856304414257, 5.667138817245209, 6.118599501188024, 6.484380418124123, 6.923235282575692, 7.506056856517067, 8.463148055302222, 8.769866460272373, 9.219221499833885, 9.825307108871997, 10.27813079963287, 10.56254586829116, 11.13360929741135, 11.85090623465159, 12.23478773902087, 12.67659622812228, 13.12202350367116
