| L(s) = 1 | + 2·7-s − 11-s + 13-s + 4·17-s − 19-s − 3·23-s + 5·29-s + 5·31-s + 2·37-s − 2·41-s + 5·43-s + 12·47-s − 3·49-s − 8·53-s − 2·59-s − 2·61-s + 2·67-s + 15·71-s − 6·73-s − 2·77-s − 8·79-s − 3·83-s − 17·89-s + 2·91-s − 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.625·23-s + 0.928·29-s + 0.898·31-s + 0.328·37-s − 0.312·41-s + 0.762·43-s + 1.75·47-s − 3/7·49-s − 1.09·53-s − 0.260·59-s − 0.256·61-s + 0.244·67-s + 1.78·71-s − 0.702·73-s − 0.227·77-s − 0.900·79-s − 0.329·83-s − 1.80·89-s + 0.209·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.74589109732361, −13.01356586708141, −12.35747446399471, −12.27763246023529, −11.63132743582810, −11.07633648932742, −10.77447709021509, −10.14769565917604, −9.790575255662978, −9.254145106445786, −8.515625296925391, −8.215055487269333, −7.849033241614632, −7.241929881552122, −6.722420050436164, −6.007188839033760, −5.729357947357349, −5.055158637439616, −4.538656792167340, −4.089941240571492, −3.403001571518404, −2.741526483436033, −2.274228984945277, −1.391718310709525, −1.007630634724486, 0,
1.007630634724486, 1.391718310709525, 2.274228984945277, 2.741526483436033, 3.403001571518404, 4.089941240571492, 4.538656792167340, 5.055158637439616, 5.729357947357349, 6.007188839033760, 6.722420050436164, 7.241929881552122, 7.849033241614632, 8.215055487269333, 8.515625296925391, 9.254145106445786, 9.790575255662978, 10.14769565917604, 10.77447709021509, 11.07633648932742, 11.63132743582810, 12.27763246023529, 12.35747446399471, 13.01356586708141, 13.74589109732361
