| L(s) = 1 | + 2-s + 4-s − 0.854·5-s + 0.727·7-s + 8-s − 0.854·10-s + 4.77·11-s + 0.670·13-s + 0.727·14-s + 16-s − 0.231·17-s − 5.18·19-s − 0.854·20-s + 4.77·22-s − 7.77·23-s − 4.27·25-s + 0.670·26-s + 0.727·28-s − 7.62·29-s − 3.13·31-s + 32-s − 0.231·34-s − 0.621·35-s − 2.35·37-s − 5.18·38-s − 0.854·40-s + 0.350·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.382·5-s + 0.274·7-s + 0.353·8-s − 0.270·10-s + 1.44·11-s + 0.185·13-s + 0.194·14-s + 0.250·16-s − 0.0561·17-s − 1.18·19-s − 0.191·20-s + 1.01·22-s − 1.62·23-s − 0.854·25-s + 0.131·26-s + 0.137·28-s − 1.41·29-s − 0.563·31-s + 0.176·32-s − 0.0397·34-s − 0.104·35-s − 0.387·37-s − 0.840·38-s − 0.135·40-s + 0.0547·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 0.854T + 5T^{2} \) |
| 7 | \( 1 - 0.727T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 0.670T + 13T^{2} \) |
| 17 | \( 1 + 0.231T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 + 7.62T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 - 0.350T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 7.38T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 - 0.603T + 83T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 + 4.56T + 97T^{2} \) |
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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
−7.34608138748023795065834312909, −6.68473816163114357323748994220, −6.08443036611326713017071036598, −5.44905633907424047514353395974, −4.42599757099661511998237368298, −3.93747499532285531115788733476, −3.46279417612143003195773554899, −2.10456334542837513793378179430, −1.60299192201221603361601808228, 0,
1.60299192201221603361601808228, 2.10456334542837513793378179430, 3.46279417612143003195773554899, 3.93747499532285531115788733476, 4.42599757099661511998237368298, 5.44905633907424047514353395974, 6.08443036611326713017071036598, 6.68473816163114357323748994220, 7.34608138748023795065834312909
