| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s − 19-s + 21-s + 25-s + 27-s + 31-s − 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s − 57-s − 59-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s − 19-s + 21-s + 25-s + 27-s + 31-s − 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s − 57-s − 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.454807590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.454807590\) |
| \(L(1)\) |
\(\approx\) |
\(1.485506399\) |
| \(L(1)\) |
\(\approx\) |
\(1.485506399\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.12117298895402161238302924644, −18.77520949648345930635840687142, −18.214030682439480479595637292924, −17.194148432373403080752923122881, −16.25312996222773223317780347355, −15.624501016091290601383112182127, −15.046280472152532651258249782040, −14.49517291938984867056185384618, −13.65804968641239437788135283603, −12.99679982324745947033844710392, −12.205302849540163942944768180536, −11.40236487566133503806909934529, −10.645556646209376843699625981797, −10.04275719274133295037212630870, −8.818815402443816030651617504435, −8.28728503568267950805517713019, −7.89831098647238339952940957013, −7.19388486188989761795008992869, −6.12320297330885256876707468540, −4.96280721894586138749738122157, −4.37030651172204716360751351334, −3.545780148787458830614166856524, −2.83090415328324623849431433232, −1.850106516063915121338378328612, −0.91125877778430314079910888424,
0.91125877778430314079910888424, 1.850106516063915121338378328612, 2.83090415328324623849431433232, 3.545780148787458830614166856524, 4.37030651172204716360751351334, 4.96280721894586138749738122157, 6.12320297330885256876707468540, 7.19388486188989761795008992869, 7.89831098647238339952940957013, 8.28728503568267950805517713019, 8.818815402443816030651617504435, 10.04275719274133295037212630870, 10.645556646209376843699625981797, 11.40236487566133503806909934529, 12.205302849540163942944768180536, 12.99679982324745947033844710392, 13.65804968641239437788135283603, 14.49517291938984867056185384618, 15.046280472152532651258249782040, 15.624501016091290601383112182127, 16.25312996222773223317780347355, 17.194148432373403080752923122881, 18.214030682439480479595637292924, 18.77520949648345930635840687142, 19.12117298895402161238302924644
