| L(s) = 1 | + 2.81·3-s − 3.15·5-s − 0.839·7-s + 4.90·9-s − 1.24·11-s + 1.89·13-s − 8.86·15-s − 17-s − 0.743·19-s − 2.36·21-s + 2.07·23-s + 4.93·25-s + 5.35·27-s + 3.56·29-s − 4.66·31-s − 3.50·33-s + 2.64·35-s + 2.24·37-s + 5.31·39-s − 9.64·41-s − 6.02·43-s − 15.4·45-s + 10.0·47-s − 6.29·49-s − 2.81·51-s − 8.14·53-s + 3.93·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s − 1.40·5-s − 0.317·7-s + 1.63·9-s − 0.376·11-s + 0.524·13-s − 2.28·15-s − 0.242·17-s − 0.170·19-s − 0.515·21-s + 0.432·23-s + 0.986·25-s + 1.03·27-s + 0.662·29-s − 0.837·31-s − 0.610·33-s + 0.447·35-s + 0.368·37-s + 0.851·39-s − 1.50·41-s − 0.919·43-s − 2.30·45-s + 1.46·47-s − 0.899·49-s − 0.393·51-s − 1.11·53-s + 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 0.839T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 - 1.89T + 13T^{2} \) |
| 19 | \( 1 + 0.743T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 61 | \( 1 - 7.48T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 8.72T + 73T^{2} \) |
| 79 | \( 1 + 0.461T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 0.141T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−7.67059207768403380691778385836, −7.08316714879747848982080147371, −6.38462088425127452387594273979, −5.15953276146142592688276457138, −4.34858127977372579128962591336, −3.68801376696577576933063585618, −3.23156977691824687825623189061, −2.49255852573541391319438393919, −1.41550131828212464343142688538, 0,
1.41550131828212464343142688538, 2.49255852573541391319438393919, 3.23156977691824687825623189061, 3.68801376696577576933063585618, 4.34858127977372579128962591336, 5.15953276146142592688276457138, 6.38462088425127452387594273979, 7.08316714879747848982080147371, 7.67059207768403380691778385836
