| L(s) = 1 | − 2·4-s − 7-s + 6·13-s + 4·16-s + 5·17-s − 19-s + 3·23-s + 2·28-s − 3·29-s + 2·31-s + 10·37-s − 4·41-s − 9·43-s + 8·47-s + 49-s − 12·52-s + 53-s + 5·59-s + 7·61-s − 8·64-s + 2·67-s − 10·68-s + 2·73-s + 2·76-s − 12·79-s − 9·83-s + 9·89-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 1.66·13-s + 16-s + 1.21·17-s − 0.229·19-s + 0.625·23-s + 0.377·28-s − 0.557·29-s + 0.359·31-s + 1.64·37-s − 0.624·41-s − 1.37·43-s + 1.16·47-s + 1/7·49-s − 1.66·52-s + 0.137·53-s + 0.650·59-s + 0.896·61-s − 64-s + 0.244·67-s − 1.21·68-s + 0.234·73-s + 0.229·76-s − 1.35·79-s − 0.987·83-s + 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−13.30174260439520, −12.97198900421371, −12.62883952845099, −11.86669082662802, −11.55052960469882, −10.97241889165558, −10.37806263755696, −10.05403130081970, −9.541422925665090, −9.048623332485555, −8.614348794719563, −8.166443356255282, −7.792449961470698, −7.049582642576855, −6.565950469894578, −5.800737281709303, −5.720812227628811, −5.040064222173816, −4.376956737499824, −3.905001593206676, −3.416064669570647, −3.019228440877724, −2.128946928861301, −1.125814559219706, −1.005189734177875, 0,
1.005189734177875, 1.125814559219706, 2.128946928861301, 3.019228440877724, 3.416064669570647, 3.905001593206676, 4.376956737499824, 5.040064222173816, 5.720812227628811, 5.800737281709303, 6.565950469894578, 7.049582642576855, 7.792449961470698, 8.166443356255282, 8.614348794719563, 9.048623332485555, 9.541422925665090, 10.05403130081970, 10.37806263755696, 10.97241889165558, 11.55052960469882, 11.86669082662802, 12.62883952845099, 12.97198900421371, 13.30174260439520
