| L(s) = 1 | + 1.69·2-s + 0.888·4-s − 3.69·5-s − 1.58·7-s − 1.88·8-s − 6.28·10-s − 4.58·11-s + 1.58·13-s − 2.69·14-s − 4.98·16-s + 5.57·17-s − 19-s − 3.28·20-s − 7.79·22-s − 9.09·23-s + 8.68·25-s + 2.69·26-s − 1.41·28-s + 7.65·29-s − 2.58·31-s − 4.69·32-s + 9.47·34-s + 5.87·35-s + 0.287·37-s − 1.69·38-s + 6.98·40-s − 3.69·41-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.444·4-s − 1.65·5-s − 0.600·7-s − 0.667·8-s − 1.98·10-s − 1.38·11-s + 0.440·13-s − 0.721·14-s − 1.24·16-s + 1.35·17-s − 0.229·19-s − 0.735·20-s − 1.66·22-s − 1.89·23-s + 1.73·25-s + 0.529·26-s − 0.266·28-s + 1.42·29-s − 0.464·31-s − 0.830·32-s + 1.62·34-s + 0.993·35-s + 0.0473·37-s − 0.275·38-s + 1.10·40-s − 0.577·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 0.287T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 0.189T + 61T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 6.21T + 71T^{2} \) |
| 73 | \( 1 + 1.22T + 73T^{2} \) |
| 79 | \( 1 + 5.90T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−10.70141714097478987142796736174, −9.706618595244469184561081209431, −8.246580173661677262153673240786, −7.85536689437410125201983055253, −6.57286336045241045337877697263, −5.56323591868199539041145557885, −4.54421227183244497240046500964, −3.65933730044698826430994695342, −2.92702245238478991078271680579, 0,
2.92702245238478991078271680579, 3.65933730044698826430994695342, 4.54421227183244497240046500964, 5.56323591868199539041145557885, 6.57286336045241045337877697263, 7.85536689437410125201983055253, 8.246580173661677262153673240786, 9.706618595244469184561081209431, 10.70141714097478987142796736174
