| L(s) = 1 | + 5-s + 3.20·7-s − 5.11·11-s + 0.597·13-s + 5.98·17-s − 3.11·19-s + 23-s + 25-s + 4.64·29-s + 3.39·31-s + 3.20·35-s − 6.51·37-s + 11.4·41-s − 3.71·43-s − 1.80·47-s + 3.28·49-s + 8.13·53-s − 5.11·55-s − 3.80·59-s − 7.07·61-s + 0.597·65-s + 0.0987·67-s + 9.51·71-s − 7.41·73-s − 16.3·77-s − 5.96·79-s − 4.55·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.21·7-s − 1.54·11-s + 0.165·13-s + 1.45·17-s − 0.714·19-s + 0.208·23-s + 0.200·25-s + 0.862·29-s + 0.608·31-s + 0.542·35-s − 1.07·37-s + 1.79·41-s − 0.565·43-s − 0.263·47-s + 0.468·49-s + 1.11·53-s − 0.689·55-s − 0.495·59-s − 0.905·61-s + 0.0741·65-s + 0.0120·67-s + 1.12·71-s − 0.867·73-s − 1.86·77-s − 0.670·79-s − 0.499·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.560510131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.560510131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 13 | \( 1 - 0.597T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 29 | \( 1 - 4.64T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 - 8.13T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 0.0987T + 67T^{2} \) |
| 71 | \( 1 - 9.51T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 + 4.55T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.76798517180904920059201543929, −7.38522821028786802220900900463, −6.28314200368632858176651838505, −5.65371702739633544924210660402, −5.00144254226475871509607021802, −4.53832190395563445610038810743, −3.37654742283708812921054201548, −2.58307139298571321093782278126, −1.80220197294988521264035880407, −0.805474589480996308870296806476,
0.805474589480996308870296806476, 1.80220197294988521264035880407, 2.58307139298571321093782278126, 3.37654742283708812921054201548, 4.53832190395563445610038810743, 5.00144254226475871509607021802, 5.65371702739633544924210660402, 6.28314200368632858176651838505, 7.38522821028786802220900900463, 7.76798517180904920059201543929
