| L(s) = 1 | − 2.52·2-s − 2.42·3-s + 4.39·4-s + 5-s + 6.13·6-s + 0.256·7-s − 6.06·8-s + 2.87·9-s − 2.52·10-s + 0.560·11-s − 10.6·12-s − 5.31·13-s − 0.649·14-s − 2.42·15-s + 6.54·16-s − 6.32·17-s − 7.28·18-s + 0.712·19-s + 4.39·20-s − 0.623·21-s − 1.41·22-s + 6.12·23-s + 14.7·24-s + 25-s + 13.4·26-s + 0.293·27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.39·3-s + 2.19·4-s + 0.447·5-s + 2.50·6-s + 0.0971·7-s − 2.14·8-s + 0.959·9-s − 0.799·10-s + 0.169·11-s − 3.07·12-s − 1.47·13-s − 0.173·14-s − 0.626·15-s + 1.63·16-s − 1.53·17-s − 1.71·18-s + 0.163·19-s + 0.983·20-s − 0.135·21-s − 0.302·22-s + 1.27·23-s + 3.00·24-s + 0.200·25-s + 2.63·26-s + 0.0564·27-s + 0.213·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1621795677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1621795677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 + 2.42T + 3T^{2} \) |
| 7 | \( 1 - 0.256T + 7T^{2} \) |
| 11 | \( 1 - 0.560T + 11T^{2} \) |
| 13 | \( 1 + 5.31T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 0.712T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 0.740T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 + 0.700T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−7.81115760454026202473511126785, −7.02626461365153022598280855238, −6.70115686040854860300941382623, −6.12607496510016323924970359042, −5.00724404756386591763727841533, −4.80221152624219813188469411924, −3.13698409330858802999078983555, −2.16829090792103188349807460652, −1.44494171425660561201866115177, −0.28860628993724328385946112248,
0.28860628993724328385946112248, 1.44494171425660561201866115177, 2.16829090792103188349807460652, 3.13698409330858802999078983555, 4.80221152624219813188469411924, 5.00724404756386591763727841533, 6.12607496510016323924970359042, 6.70115686040854860300941382623, 7.02626461365153022598280855238, 7.81115760454026202473511126785
