| L(s) = 1 | − 2.61·2-s − 3-s + 4.85·4-s + 3·5-s + 2.61·6-s − 7.47·8-s + 9-s − 7.85·10-s + 3.47·11-s − 4.85·12-s + 6.70·13-s − 3·15-s + 9.85·16-s + 2.61·17-s − 2.61·18-s + 5.85·19-s + 14.5·20-s − 9.09·22-s − 1.38·23-s + 7.47·24-s + 4·25-s − 17.5·26-s − 27-s − 4.38·29-s + 7.85·30-s + 2.38·31-s − 10.8·32-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.42·4-s + 1.34·5-s + 1.06·6-s − 2.64·8-s + 0.333·9-s − 2.48·10-s + 1.04·11-s − 1.40·12-s + 1.86·13-s − 0.774·15-s + 2.46·16-s + 0.634·17-s − 0.617·18-s + 1.34·19-s + 3.25·20-s − 1.93·22-s − 0.288·23-s + 1.52·24-s + 0.800·25-s − 3.44·26-s − 0.192·27-s − 0.813·29-s + 1.43·30-s + 0.427·31-s − 1.91·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.411843445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411843445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 - 5.47T + 53T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 6.09T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−7.71497403740230538730075730255, −7.30978946309367927068872545662, −6.26224114105384917204530355217, −6.07621856411657874394924906588, −5.57668980936888796884375075709, −4.13174992398160337816013969668, −3.15282127242341154530069099977, −2.14378250282753644018431845573, −1.19859717345043575729236220870, −1.02989656765780724568060027345,
1.02989656765780724568060027345, 1.19859717345043575729236220870, 2.14378250282753644018431845573, 3.15282127242341154530069099977, 4.13174992398160337816013969668, 5.57668980936888796884375075709, 6.07621856411657874394924906588, 6.26224114105384917204530355217, 7.30978946309367927068872545662, 7.71497403740230538730075730255
