L(s) = 1 | + 0.488·3-s − 3.65·5-s + 7-s − 2.76·9-s + 11-s + 13-s − 1.78·15-s + 1.65·17-s − 0.616·19-s + 0.488·21-s + 4.52·23-s + 8.37·25-s − 2.81·27-s + 3.33·29-s + 3.54·31-s + 0.488·33-s − 3.65·35-s − 9.82·37-s + 0.488·39-s + 1.93·41-s + 0.00506·43-s + 10.0·45-s + 4.15·47-s + 49-s + 0.808·51-s + 3.08·53-s − 3.65·55-s + ⋯ |
L(s) = 1 | + 0.281·3-s − 1.63·5-s + 0.377·7-s − 0.920·9-s + 0.301·11-s + 0.277·13-s − 0.460·15-s + 0.401·17-s − 0.141·19-s + 0.106·21-s + 0.943·23-s + 1.67·25-s − 0.541·27-s + 0.618·29-s + 0.637·31-s + 0.0849·33-s − 0.618·35-s − 1.61·37-s + 0.0781·39-s + 0.302·41-s + 0.000772·43-s + 1.50·45-s + 0.605·47-s + 0.142·49-s + 0.113·51-s + 0.423·53-s − 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.488T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 0.616T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 1.93T + 41T^{2} \) |
| 43 | \( 1 - 0.00506T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 - 1.62T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 + 2.98T + 83T^{2} \) |
| 89 | \( 1 + 9.40T + 89T^{2} \) |
| 97 | \( 1 + 0.425T + 97T^{2} \) |
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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
−8.112670851926190485735173727452, −7.50796568544953921956542030756, −6.80562267735997535951527864877, −5.84050222686778956114995703351, −4.91242241679693865770921964797, −4.20422099648493053318122373227, −3.37949161426529445494788376707, −2.78204687161298935847381370150, −1.26771153294288235159964312295, 0,
1.26771153294288235159964312295, 2.78204687161298935847381370150, 3.37949161426529445494788376707, 4.20422099648493053318122373227, 4.91242241679693865770921964797, 5.84050222686778956114995703351, 6.80562267735997535951527864877, 7.50796568544953921956542030756, 8.112670851926190485735173727452