| L(s) = 1 | + 0.583·2-s − 3-s − 1.65·4-s − 2.20·5-s − 0.583·6-s − 7-s − 2.13·8-s + 9-s − 1.28·10-s + 0.238·11-s + 1.65·12-s + 2.66·13-s − 0.583·14-s + 2.20·15-s + 2.07·16-s − 5.07·17-s + 0.583·18-s + 1.66·19-s + 3.66·20-s + 21-s + 0.138·22-s − 1.34·23-s + 2.13·24-s − 0.124·25-s + 1.55·26-s − 27-s + 1.65·28-s + ⋯ |
| L(s) = 1 | + 0.412·2-s − 0.577·3-s − 0.829·4-s − 0.987·5-s − 0.238·6-s − 0.377·7-s − 0.754·8-s + 0.333·9-s − 0.407·10-s + 0.0718·11-s + 0.479·12-s + 0.739·13-s − 0.155·14-s + 0.570·15-s + 0.518·16-s − 1.22·17-s + 0.137·18-s + 0.381·19-s + 0.819·20-s + 0.218·21-s + 0.0296·22-s − 0.279·23-s + 0.435·24-s − 0.0249·25-s + 0.304·26-s − 0.192·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4392115915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4392115915\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.583T + 2T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 11 | \( 1 - 0.238T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 + 5.07T + 17T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 + 9.43T + 41T^{2} \) |
| 43 | \( 1 - 2.66T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 3.21T + 73T^{2} \) |
| 79 | \( 1 + 6.46T + 79T^{2} \) |
| 83 | \( 1 - 4.79T + 83T^{2} \) |
| 89 | \( 1 + 9.63T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.924562486177632426775258880060, −6.98144277877338221474099627743, −6.38713919611279867590225208100, −5.71372303036985815499057831007, −4.84576968776460382096330188152, −4.37158557863364132322728628253, −3.64611209215711650158241197176, −3.07673754304769405420398897931, −1.60169731677969047293762463622, −0.32242376642068424845111144552,
0.32242376642068424845111144552, 1.60169731677969047293762463622, 3.07673754304769405420398897931, 3.64611209215711650158241197176, 4.37158557863364132322728628253, 4.84576968776460382096330188152, 5.71372303036985815499057831007, 6.38713919611279867590225208100, 6.98144277877338221474099627743, 7.924562486177632426775258880060
