| L(s) = 1 | − 1.44·2-s + 0.0986·4-s − 2.08·5-s + 4.31·7-s + 2.75·8-s + 3.02·10-s − 0.927·11-s + 0.332·13-s − 6.25·14-s − 4.18·16-s − 6.67·17-s − 4.65·19-s − 0.205·20-s + 1.34·22-s + 23-s − 0.641·25-s − 0.482·26-s + 0.425·28-s + 29-s + 2.69·31-s + 0.557·32-s + 9.66·34-s − 9.01·35-s − 6.22·37-s + 6.74·38-s − 5.75·40-s + 2.85·41-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0493·4-s − 0.933·5-s + 1.63·7-s + 0.973·8-s + 0.956·10-s − 0.279·11-s + 0.0922·13-s − 1.67·14-s − 1.04·16-s − 1.61·17-s − 1.06·19-s − 0.0460·20-s + 0.286·22-s + 0.208·23-s − 0.128·25-s − 0.0945·26-s + 0.0804·28-s + 0.185·29-s + 0.484·31-s + 0.0985·32-s + 1.65·34-s − 1.52·35-s − 1.02·37-s + 1.09·38-s − 0.909·40-s + 0.446·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7054945972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7054945972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 + 0.927T + 11T^{2} \) |
| 13 | \( 1 - 0.332T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−8.310383874126665923496094421769, −7.50941510474656792567626306297, −7.14751584150169407206508321056, −6.04846865849568308695546942226, −4.90215477516155055816822196265, −4.49296315838187934060317590186, −3.91264756803471319023623370492, −2.39925405246360151629287783152, −1.68216329232494230747754909904, −0.52675487589962159543976983720,
0.52675487589962159543976983720, 1.68216329232494230747754909904, 2.39925405246360151629287783152, 3.91264756803471319023623370492, 4.49296315838187934060317590186, 4.90215477516155055816822196265, 6.04846865849568308695546942226, 7.14751584150169407206508321056, 7.50941510474656792567626306297, 8.310383874126665923496094421769
