| L(s) = 1 | − 3-s + 9-s + 2·11-s + 6·13-s + 4·19-s − 27-s + 2·31-s − 2·33-s + 4·37-s − 6·39-s + 2·41-s + 10·43-s − 6·47-s + 14·53-s − 4·57-s + 4·59-s + 2·61-s − 2·67-s + 6·73-s − 16·79-s + 81-s + 8·83-s + 10·89-s − 2·93-s − 14·97-s + 2·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.66·13-s + 0.917·19-s − 0.192·27-s + 0.359·31-s − 0.348·33-s + 0.657·37-s − 0.960·39-s + 0.312·41-s + 1.52·43-s − 0.875·47-s + 1.92·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.244·67-s + 0.702·73-s − 1.80·79-s + 1/9·81-s + 0.878·83-s + 1.05·89-s − 0.207·93-s − 1.42·97-s + 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.905389228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.905389228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.23702650185996, −13.84984510459083, −13.18058701735699, −12.99673311411321, −12.14850318388731, −11.73810105374879, −11.37576116782399, −10.75966937807991, −10.44310192487455, −9.587922084898197, −9.338667566193183, −8.581381462570714, −8.204800137227178, −7.456401861961949, −6.929164389537017, −6.377970050805205, −5.782199005228551, −5.523751274291334, −4.599875107323047, −4.078451481986311, −3.561128424845090, −2.851814659172682, −1.953516896636226, −1.119312418172693, −0.7341909957071104,
0.7341909957071104, 1.119312418172693, 1.953516896636226, 2.851814659172682, 3.561128424845090, 4.078451481986311, 4.599875107323047, 5.523751274291334, 5.782199005228551, 6.377970050805205, 6.929164389537017, 7.456401861961949, 8.204800137227178, 8.581381462570714, 9.338667566193183, 9.587922084898197, 10.44310192487455, 10.75966937807991, 11.37576116782399, 11.73810105374879, 12.14850318388731, 12.99673311411321, 13.18058701735699, 13.84984510459083, 14.23702650185996
