L(s) = 1 | + 2-s − 1.22·3-s + 4-s − 2.52·5-s − 1.22·6-s + 3.13·7-s + 8-s − 1.48·9-s − 2.52·10-s + 2.37·11-s − 1.22·12-s + 2.61·13-s + 3.13·14-s + 3.10·15-s + 16-s + 6.05·17-s − 1.48·18-s + 19-s − 2.52·20-s − 3.85·21-s + 2.37·22-s + 5.67·23-s − 1.22·24-s + 1.37·25-s + 2.61·26-s + 5.51·27-s + 3.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.709·3-s + 0.5·4-s − 1.12·5-s − 0.501·6-s + 1.18·7-s + 0.353·8-s − 0.496·9-s − 0.798·10-s + 0.716·11-s − 0.354·12-s + 0.725·13-s + 0.838·14-s + 0.801·15-s + 0.250·16-s + 1.46·17-s − 0.350·18-s + 0.229·19-s − 0.564·20-s − 0.841·21-s + 0.506·22-s + 1.18·23-s − 0.250·24-s + 0.275·25-s + 0.513·26-s + 1.06·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637635063\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637635063\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 - 1.51T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 - 6.27T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 8.25T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.83121961305033556480859919418, −7.09588637447067004826526610687, −6.33649156722969454696990274418, −5.56276840903480166657833870885, −5.08429707493133878686148541773, −4.35451826763289433238979804448, −3.61859765271631244565967419258, −3.01012237295572542795609471669, −1.60432571993011190294230862211, −0.808354502194364319217541307471,
0.808354502194364319217541307471, 1.60432571993011190294230862211, 3.01012237295572542795609471669, 3.61859765271631244565967419258, 4.35451826763289433238979804448, 5.08429707493133878686148541773, 5.56276840903480166657833870885, 6.33649156722969454696990274418, 7.09588637447067004826526610687, 7.83121961305033556480859919418