| L(s) = 1 | + 2-s − 0.873·3-s + 4-s + 3.87·5-s − 0.873·6-s + 2.67·7-s + 8-s − 2.23·9-s + 3.87·10-s − 2.90·11-s − 0.873·12-s + 1.11·13-s + 2.67·14-s − 3.38·15-s + 16-s + 17-s − 2.23·18-s + 0.541·19-s + 3.87·20-s − 2.33·21-s − 2.90·22-s + 5.12·23-s − 0.873·24-s + 9.99·25-s + 1.11·26-s + 4.57·27-s + 2.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.504·3-s + 0.5·4-s + 1.73·5-s − 0.356·6-s + 1.01·7-s + 0.353·8-s − 0.745·9-s + 1.22·10-s − 0.876·11-s − 0.252·12-s + 0.310·13-s + 0.714·14-s − 0.872·15-s + 0.250·16-s + 0.242·17-s − 0.527·18-s + 0.124·19-s + 0.865·20-s − 0.509·21-s − 0.620·22-s + 1.06·23-s − 0.178·24-s + 1.99·25-s + 0.219·26-s + 0.880·27-s + 0.505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.407136585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.407136585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.873T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 19 | \( 1 - 0.541T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 0.720T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 6.77T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 + 0.143T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 5.34T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 - 1.30T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 9.66T + 89T^{2} \) |
| 97 | \( 1 - 7.56T + 97T^{2} \) |
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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
−9.172394360045327636710875485607, −8.410799903635217272128079018599, −7.44354690748366747465389615613, −6.43782664703125777001213000202, −5.76237177769614248386505017864, −5.25530636777461471356470084007, −4.68676872298449392660904660199, −3.08661854251317501826449044305, −2.30440868235115838504251656538, −1.26845050633358815433344640171,
1.26845050633358815433344640171, 2.30440868235115838504251656538, 3.08661854251317501826449044305, 4.68676872298449392660904660199, 5.25530636777461471356470084007, 5.76237177769614248386505017864, 6.43782664703125777001213000202, 7.44354690748366747465389615613, 8.410799903635217272128079018599, 9.172394360045327636710875485607
