| L(s) = 1 | + 2-s + 4-s + 3.87·5-s + 1.22·7-s + 8-s + 3.87·10-s + 2.12·11-s − 0.490·13-s + 1.22·14-s + 16-s + 5.53·17-s + 3.87·20-s + 2.12·22-s + 8.94·23-s + 10.0·25-s − 0.490·26-s + 1.22·28-s − 8.47·29-s − 2.41·31-s + 32-s + 5.53·34-s + 4.75·35-s − 1.69·37-s + 3.87·40-s + 1.59·41-s − 6.63·43-s + 2.12·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.73·5-s + 0.463·7-s + 0.353·8-s + 1.22·10-s + 0.639·11-s − 0.135·13-s + 0.327·14-s + 0.250·16-s + 1.34·17-s + 0.867·20-s + 0.452·22-s + 1.86·23-s + 2.00·25-s − 0.0961·26-s + 0.231·28-s − 1.57·29-s − 0.433·31-s + 0.176·32-s + 0.948·34-s + 0.804·35-s − 0.278·37-s + 0.613·40-s + 0.249·41-s − 1.01·43-s + 0.319·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.569004915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.569004915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 0.490T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 - 1.59T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 0.0418T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.83902365861263186348353012066, −7.09967870679954341238214048674, −6.48488162446852756865546962942, −5.64704653108366398115178661069, −5.35784637712948465808897734787, −4.61687633458230842254207904549, −3.49356617298099672000088178568, −2.81764246259747334122041258119, −1.76914424543634750095115935828, −1.28528908376966984438506609233,
1.28528908376966984438506609233, 1.76914424543634750095115935828, 2.81764246259747334122041258119, 3.49356617298099672000088178568, 4.61687633458230842254207904549, 5.35784637712948465808897734787, 5.64704653108366398115178661069, 6.48488162446852756865546962942, 7.09967870679954341238214048674, 7.83902365861263186348353012066
