| L(s) = 1 | − 1.49·3-s + 0.0991·5-s + 1.31·7-s − 0.773·9-s − 2.16·11-s − 3.30·13-s − 0.147·15-s − 0.622·17-s + 7.82·19-s − 1.96·21-s + 4.06·23-s − 4.99·25-s + 5.63·27-s + 0.0908·29-s + 2.93·31-s + 3.23·33-s + 0.130·35-s + 3.54·37-s + 4.92·39-s + 1.35·41-s − 2.82·43-s − 0.0766·45-s − 12.2·47-s − 5.26·49-s + 0.928·51-s + 4.57·53-s − 0.215·55-s + ⋯ |
| L(s) = 1 | − 0.861·3-s + 0.0443·5-s + 0.497·7-s − 0.257·9-s − 0.654·11-s − 0.915·13-s − 0.0381·15-s − 0.150·17-s + 1.79·19-s − 0.428·21-s + 0.847·23-s − 0.998·25-s + 1.08·27-s + 0.0168·29-s + 0.526·31-s + 0.563·33-s + 0.0220·35-s + 0.583·37-s + 0.788·39-s + 0.211·41-s − 0.430·43-s − 0.0114·45-s − 1.78·47-s − 0.752·49-s + 0.130·51-s + 0.627·53-s − 0.0289·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 0.0991T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 0.622T + 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 - 0.0908T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 - 4.01T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 - 0.767T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 + 9.59T + 79T^{2} \) |
| 83 | \( 1 + 0.137T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 4.66T + 97T^{2} \) |
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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
−7.923415613559789119457359250627, −7.40686860241791668191038398686, −6.55399844554265018470007133754, −5.68947286300171353641847661248, −5.12241859784295149761201483280, −4.63749340292819025374388115351, −3.30346402801647674540511447943, −2.51950151747918728674352320091, −1.23913746287921926948979661302, 0,
1.23913746287921926948979661302, 2.51950151747918728674352320091, 3.30346402801647674540511447943, 4.63749340292819025374388115351, 5.12241859784295149761201483280, 5.68947286300171353641847661248, 6.55399844554265018470007133754, 7.40686860241791668191038398686, 7.923415613559789119457359250627
