| L(s) = 1 | + 2.50·2-s + 4.26·4-s + 5-s − 2.88·7-s + 5.68·8-s + 2.50·10-s − 3.10·11-s − 0.735·13-s − 7.21·14-s + 5.68·16-s + 7.46·17-s + 6.34·19-s + 4.26·20-s − 7.77·22-s + 7.29·23-s + 25-s − 1.84·26-s − 12.3·28-s + 4.07·29-s − 0.449·31-s + 2.87·32-s + 18.6·34-s − 2.88·35-s + 7.81·37-s + 15.8·38-s + 5.68·40-s − 4.57·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.13·4-s + 0.447·5-s − 1.08·7-s + 2.00·8-s + 0.791·10-s − 0.936·11-s − 0.204·13-s − 1.92·14-s + 1.42·16-s + 1.81·17-s + 1.45·19-s + 0.954·20-s − 1.65·22-s + 1.52·23-s + 0.200·25-s − 0.361·26-s − 2.32·28-s + 0.756·29-s − 0.0807·31-s + 0.508·32-s + 3.20·34-s − 0.487·35-s + 1.28·37-s + 2.57·38-s + 0.898·40-s − 0.714·41-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
| \(L(1)\) |
\(\approx\) |
\(5.962128459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.962128459\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.50T + 2T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 + 0.735T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 0.449T + 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 + 4.57T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 0.356T + 47T^{2} \) |
| 53 | \( 1 + 0.999T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.172888411661460705707858764811, −7.31985011460728733034343609544, −6.80256213969277459075233037180, −5.93761536346819186477725932127, −5.35056095291897637929299614966, −4.94094710316947763152911162034, −3.71427282720827196593237508519, −3.02665006054815970081566200108, −2.66919579825721578974288231959, −1.13548182515048162070673923011,
1.13548182515048162070673923011, 2.66919579825721578974288231959, 3.02665006054815970081566200108, 3.71427282720827196593237508519, 4.94094710316947763152911162034, 5.35056095291897637929299614966, 5.93761536346819186477725932127, 6.80256213969277459075233037180, 7.31985011460728733034343609544, 8.172888411661460705707858764811
