| L(s) = 1 | + 2·17-s − 19-s − 6·23-s − 5·25-s + 10·29-s + 10·31-s − 8·37-s − 6·41-s + 4·43-s + 6·47-s − 7·49-s − 6·53-s + 12·59-s + 10·61-s − 8·67-s − 8·71-s − 2·73-s + 6·79-s + 12·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.485·17-s − 0.229·19-s − 1.25·23-s − 25-s + 1.85·29-s + 1.79·31-s − 1.31·37-s − 0.937·41-s + 0.609·43-s + 0.875·47-s − 49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.977·67-s − 0.949·71-s − 0.234·73-s + 0.675·79-s + 1.31·83-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.884445281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884445281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.32102627070502, −15.92987164074540, −15.52957639981502, −14.77210282849788, −14.09004549941620, −13.78914195492771, −13.16844356307709, −12.25386483816261, −11.99229355233021, −11.49060803225463, −10.35723581576424, −10.28161023816329, −9.601640412230387, −8.711909927270615, −8.204570474801078, −7.722739354037518, −6.769830822050218, −6.322981755927120, −5.591536835966850, −4.820295767977022, −4.158386441937854, −3.383165116866782, −2.558728900520434, −1.715030714216082, −0.6392624300461031,
0.6392624300461031, 1.715030714216082, 2.558728900520434, 3.383165116866782, 4.158386441937854, 4.820295767977022, 5.591536835966850, 6.322981755927120, 6.769830822050218, 7.722739354037518, 8.204570474801078, 8.711909927270615, 9.601640412230387, 10.28161023816329, 10.35723581576424, 11.49060803225463, 11.99229355233021, 12.25386483816261, 13.16844356307709, 13.78914195492771, 14.09004549941620, 14.77210282849788, 15.52957639981502, 15.92987164074540, 16.32102627070502
