| L(s) = 1 | − 1.99·2-s + 1.99·4-s − 1.55·5-s − 7-s + 0.0158·8-s + 3.09·10-s − 5.13·11-s + 3.23·13-s + 1.99·14-s − 4.01·16-s + 3.98·17-s + 2.86·19-s − 3.08·20-s + 10.2·22-s − 1.67·23-s − 2.59·25-s − 6.46·26-s − 1.99·28-s − 7.42·29-s + 0.815·31-s + 7.99·32-s − 7.95·34-s + 1.55·35-s + 0.873·37-s − 5.73·38-s − 0.0245·40-s − 5.56·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.996·4-s − 0.693·5-s − 0.377·7-s + 0.00559·8-s + 0.979·10-s − 1.54·11-s + 0.897·13-s + 0.533·14-s − 1.00·16-s + 0.965·17-s + 0.658·19-s − 0.690·20-s + 2.18·22-s − 0.350·23-s − 0.518·25-s − 1.26·26-s − 0.376·28-s − 1.37·29-s + 0.146·31-s + 1.41·32-s − 1.36·34-s + 0.262·35-s + 0.143·37-s − 0.930·38-s − 0.00388·40-s − 0.868·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4152185426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4152185426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 + 7.42T + 29T^{2} \) |
| 31 | \( 1 - 0.815T + 31T^{2} \) |
| 37 | \( 1 - 0.873T + 37T^{2} \) |
| 41 | \( 1 + 5.56T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 6.85T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 + 0.256T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 - 9.67T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 3.77T + 97T^{2} \) |
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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
−7.965735607564045827140621812893, −7.48279007821127372427385701301, −6.81984262368889454147743393830, −5.76972066706009403956798564744, −5.24689484125959984920268539129, −4.11263556606114524183539893360, −3.40519854481726558847018905388, −2.49262452383615131642627269706, −1.46025412594797667397508323548, −0.41223240978919964314969464761,
0.41223240978919964314969464761, 1.46025412594797667397508323548, 2.49262452383615131642627269706, 3.40519854481726558847018905388, 4.11263556606114524183539893360, 5.24689484125959984920268539129, 5.76972066706009403956798564744, 6.81984262368889454147743393830, 7.48279007821127372427385701301, 7.965735607564045827140621812893
