| L(s) = 1 | − 0.151·2-s − 1.97·4-s + 5-s + 2.88·7-s + 0.604·8-s − 0.151·10-s − 2.75·11-s + 2.78·13-s − 0.438·14-s + 3.86·16-s − 6.24·17-s − 0.721·19-s − 1.97·20-s + 0.418·22-s + 0.0986·23-s + 25-s − 0.423·26-s − 5.70·28-s − 6.61·29-s − 0.885·31-s − 1.79·32-s + 0.949·34-s + 2.88·35-s − 4.03·37-s + 0.109·38-s + 0.604·40-s − 9.13·41-s + ⋯ |
| L(s) = 1 | − 0.107·2-s − 0.988·4-s + 0.447·5-s + 1.09·7-s + 0.213·8-s − 0.0480·10-s − 0.830·11-s + 0.773·13-s − 0.117·14-s + 0.965·16-s − 1.51·17-s − 0.165·19-s − 0.442·20-s + 0.0892·22-s + 0.0205·23-s + 0.200·25-s − 0.0830·26-s − 1.07·28-s − 1.22·29-s − 0.159·31-s − 0.317·32-s + 0.162·34-s + 0.488·35-s − 0.662·37-s + 0.0177·38-s + 0.0955·40-s − 1.42·41-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 | \( 1 + 0.151T + 2T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 0.721T + 19T^{2} \) |
| 23 | \( 1 - 0.0986T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 0.885T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 + 0.765T + 53T^{2} \) |
| 59 | \( 1 + 2.79T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + 9.47T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.178015905718573296743677689293, −7.58496760322654945756537101388, −6.58239763411509047109482376994, −5.64382593502638586461697998359, −5.06479192218535397138150723523, −4.40128456919497806415368592408, −3.57228933279433656146416262399, −2.29161829484172047694955294984, −1.43258806366851688126908271421, 0,
1.43258806366851688126908271421, 2.29161829484172047694955294984, 3.57228933279433656146416262399, 4.40128456919497806415368592408, 5.06479192218535397138150723523, 5.64382593502638586461697998359, 6.58239763411509047109482376994, 7.58496760322654945756537101388, 8.178015905718573296743677689293
