| L(s) = 1 | − 0.644·2-s − 1.58·4-s − 5-s + 2.88·7-s + 2.31·8-s + 0.644·10-s − 5.88·11-s − 4.54·13-s − 1.85·14-s + 1.68·16-s + 2.13·17-s + 1.71·19-s + 1.58·20-s + 3.79·22-s + 7.78·23-s + 25-s + 2.93·26-s − 4.57·28-s − 4.02·29-s + 10.2·31-s − 5.70·32-s − 1.37·34-s − 2.88·35-s − 2.54·37-s − 1.10·38-s − 2.31·40-s − 0.599·41-s + ⋯ |
| L(s) = 1 | − 0.455·2-s − 0.792·4-s − 0.447·5-s + 1.09·7-s + 0.816·8-s + 0.203·10-s − 1.77·11-s − 1.26·13-s − 0.496·14-s + 0.420·16-s + 0.516·17-s + 0.393·19-s + 0.354·20-s + 0.808·22-s + 1.62·23-s + 0.200·25-s + 0.574·26-s − 0.863·28-s − 0.747·29-s + 1.83·31-s − 1.00·32-s − 0.235·34-s − 0.487·35-s − 0.418·37-s − 0.179·38-s − 0.365·40-s − 0.0935·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 0.644T + 2T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 5.88T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 3.62T + 59T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 + 3.00T + 73T^{2} \) |
| 79 | \( 1 - 4.21T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−8.017917854253482591413083699357, −7.70267817277683722419034916229, −6.96565434872272704708081194033, −5.49031014146397905735542361496, −4.88897323888192846253332578366, −4.69463770791536456101500500397, −3.33406059685106052951446235335, −2.45614056181816181833902046522, −1.17110335937212824630319454238, 0,
1.17110335937212824630319454238, 2.45614056181816181833902046522, 3.33406059685106052951446235335, 4.69463770791536456101500500397, 4.88897323888192846253332578366, 5.49031014146397905735542361496, 6.96565434872272704708081194033, 7.70267817277683722419034916229, 8.017917854253482591413083699357
