| L(s) = 1 | − 0.772·2-s + 3-s − 1.40·4-s + 5-s − 0.772·6-s + 1.62·7-s + 2.62·8-s + 9-s − 0.772·10-s − 1.40·12-s − 4.80·13-s − 1.25·14-s + 15-s + 0.772·16-s − 2.17·17-s − 0.772·18-s + 5.17·19-s − 1.40·20-s + 1.62·21-s + 4.62·23-s + 2.62·24-s + 25-s + 3.71·26-s + 27-s − 2.28·28-s − 2.45·29-s − 0.772·30-s + ⋯ |
| L(s) = 1 | − 0.546·2-s + 0.577·3-s − 0.701·4-s + 0.447·5-s − 0.315·6-s + 0.616·7-s + 0.929·8-s + 0.333·9-s − 0.244·10-s − 0.404·12-s − 1.33·13-s − 0.336·14-s + 0.258·15-s + 0.193·16-s − 0.527·17-s − 0.182·18-s + 1.18·19-s − 0.313·20-s + 0.355·21-s + 0.965·23-s + 0.536·24-s + 0.200·25-s + 0.728·26-s + 0.192·27-s − 0.432·28-s − 0.455·29-s − 0.141·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551098460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551098460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.772T + 2T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 - 4.62T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 8.88T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 - 9.72T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 4.72T + 97T^{2} \) |
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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
−9.385057483733456711529088363270, −8.613617086914132581731130873398, −7.64567541403390110697011726259, −7.40452439086391821040392787310, −6.05649699975906286599709467032, −4.86364877590769821777102640765, −4.60313745511980984402685975464, −3.19192704479617888795970635275, −2.14942376985784947632609425073, −0.941858972653989298969166853147,
0.941858972653989298969166853147, 2.14942376985784947632609425073, 3.19192704479617888795970635275, 4.60313745511980984402685975464, 4.86364877590769821777102640765, 6.05649699975906286599709467032, 7.40452439086391821040392787310, 7.64567541403390110697011726259, 8.613617086914132581731130873398, 9.385057483733456711529088363270
