| L(s) = 1 | − 0.844·2-s + 3-s − 1.28·4-s − 1.91·5-s − 0.844·6-s + 2.77·8-s + 9-s + 1.61·10-s + 2.68·11-s − 1.28·12-s + 1.87·13-s − 1.91·15-s + 0.231·16-s + 6.09·17-s − 0.844·18-s + 7.23·19-s + 2.46·20-s − 2.26·22-s − 3.42·23-s + 2.77·24-s − 1.34·25-s − 1.58·26-s + 27-s + 10.5·29-s + 1.61·30-s + 7.41·31-s − 5.74·32-s + ⋯ |
| L(s) = 1 | − 0.596·2-s + 0.577·3-s − 0.643·4-s − 0.855·5-s − 0.344·6-s + 0.981·8-s + 0.333·9-s + 0.510·10-s + 0.808·11-s − 0.371·12-s + 0.521·13-s − 0.493·15-s + 0.0579·16-s + 1.47·17-s − 0.198·18-s + 1.66·19-s + 0.550·20-s − 0.482·22-s − 0.714·23-s + 0.566·24-s − 0.268·25-s − 0.311·26-s + 0.192·27-s + 1.95·29-s + 0.294·30-s + 1.33·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610481359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610481359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.844T + 2T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 - 4.04T + 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 - 4.71T + 89T^{2} \) |
| 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
−8.103394376177089630245563788362, −7.73621394462638058324097397525, −6.93444944544294505203708400880, −6.01002523195246073173783965294, −5.02680909967248695024244838322, −4.32140186719471406825139558285, −3.58249294919847640313519381896, −3.03140742384587357961413488321, −1.44663535910573927500183546849, −0.818253767334180792473143545459,
0.818253767334180792473143545459, 1.44663535910573927500183546849, 3.03140742384587357961413488321, 3.58249294919847640313519381896, 4.32140186719471406825139558285, 5.02680909967248695024244838322, 6.01002523195246073173783965294, 6.93444944544294505203708400880, 7.73621394462638058324097397525, 8.103394376177089630245563788362
