L(s) = 1 | + 2-s + 4-s + 1.56·5-s − 7-s + 8-s + 1.56·10-s − 5.12·11-s − 3.56·13-s − 14-s + 16-s + 1.12·17-s − 5.12·19-s + 1.56·20-s − 5.12·22-s − 23-s − 2.56·25-s − 3.56·26-s − 28-s − 7.56·29-s + 3.12·31-s + 32-s + 1.12·34-s − 1.56·35-s − 1.56·37-s − 5.12·38-s + 1.56·40-s − 3.56·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.698·5-s − 0.377·7-s + 0.353·8-s + 0.493·10-s − 1.54·11-s − 0.987·13-s − 0.267·14-s + 0.250·16-s + 0.272·17-s − 1.17·19-s + 0.349·20-s − 1.09·22-s − 0.208·23-s − 0.512·25-s − 0.698·26-s − 0.188·28-s − 1.40·29-s + 0.560·31-s + 0.176·32-s + 0.192·34-s − 0.263·35-s − 0.256·37-s − 0.831·38-s + 0.246·40-s − 0.556·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 9.36T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.137428607884339326372240436451, −7.64212240913575077579625388714, −6.70499397192025405259067539151, −5.96117013321385477602204595340, −5.29672860862609689290622670443, −4.62823137949894990766042509918, −3.53861950401774083328688677691, −2.55422667344097801323175761226, −1.96806931661533329899196232018, 0,
1.96806931661533329899196232018, 2.55422667344097801323175761226, 3.53861950401774083328688677691, 4.62823137949894990766042509918, 5.29672860862609689290622670443, 5.96117013321385477602204595340, 6.70499397192025405259067539151, 7.64212240913575077579625388714, 8.137428607884339326372240436451