| L(s) = 1 | − 5-s + 7-s − 3·9-s − 4·23-s + 25-s + 4·31-s − 35-s − 10·37-s + 8·41-s + 8·43-s + 3·45-s − 4·47-s + 49-s + 6·53-s + 8·59-s + 8·61-s − 3·63-s − 4·71-s − 16·73-s + 16·79-s + 9·81-s − 8·83-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.834·23-s + 1/5·25-s + 0.718·31-s − 0.169·35-s − 1.64·37-s + 1.24·41-s + 1.21·43-s + 0.447·45-s − 0.583·47-s + 1/7·49-s + 0.824·53-s + 1.04·59-s + 1.02·61-s − 0.377·63-s − 0.474·71-s − 1.87·73-s + 1.80·79-s + 81-s − 0.878·83-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.08767026501681, −15.69917696193929, −15.01309954951960, −14.38733186681137, −14.16296906625022, −13.46393902304310, −12.81782333469182, −12.06782457350074, −11.79889446292355, −11.19096393753612, −10.62045753907701, −10.05776333622380, −9.278001838024092, −8.629103449613461, −8.281642371223213, −7.594325406972410, −7.013754686710295, −6.195547739538951, −5.643053622792291, −5.014600954087814, −4.215531916632704, −3.632967173985938, −2.771854207868852, −2.137681052648761, −1.020166568492305, 0,
1.020166568492305, 2.137681052648761, 2.771854207868852, 3.632967173985938, 4.215531916632704, 5.014600954087814, 5.643053622792291, 6.195547739538951, 7.013754686710295, 7.594325406972410, 8.281642371223213, 8.629103449613461, 9.278001838024092, 10.05776333622380, 10.62045753907701, 11.19096393753612, 11.79889446292355, 12.06782457350074, 12.81782333469182, 13.46393902304310, 14.16296906625022, 14.38733186681137, 15.01309954951960, 15.69917696193929, 16.08767026501681
