| L(s) = 1 | + 5.36·2-s − 3.86·3-s + 20.8·4-s − 16.8·5-s − 20.7·6-s + 34.1·7-s + 68.9·8-s − 12.0·9-s − 90.2·10-s − 69.6·11-s − 80.4·12-s − 86.0·13-s + 183.·14-s + 64.9·15-s + 203.·16-s − 0.387·17-s − 64.8·18-s − 17.1·19-s − 350.·20-s − 131.·21-s − 373.·22-s + 23·23-s − 266.·24-s + 157.·25-s − 462.·26-s + 150.·27-s + 711.·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.743·3-s + 2.60·4-s − 1.50·5-s − 1.41·6-s + 1.84·7-s + 3.04·8-s − 0.447·9-s − 2.85·10-s − 1.90·11-s − 1.93·12-s − 1.83·13-s + 3.50·14-s + 1.11·15-s + 3.17·16-s − 0.00553·17-s − 0.849·18-s − 0.206·19-s − 3.91·20-s − 1.37·21-s − 3.62·22-s + 0.208·23-s − 2.26·24-s + 1.26·25-s − 3.48·26-s + 1.07·27-s + 4.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.36T + 8T^{2} \) |
| 3 | \( 1 + 3.86T + 27T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 - 34.1T + 343T^{2} \) |
| 11 | \( 1 + 69.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.387T + 4.91e3T^{2} \) |
| 19 | \( 1 + 17.1T + 6.85e3T^{2} \) |
| 31 | \( 1 + 97.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 8.18T + 5.06e4T^{2} \) |
| 41 | \( 1 + 357.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 26.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 371.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 11.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 412.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 345.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 43.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 663.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 303.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 3.15T + 5.71e5T^{2} \) |
| 89 | \( 1 - 469.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 584.T + 9.12e5T^{2} \) |
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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
−10.46621144258647288430662866189, −8.073291216211193933371132679457, −7.78654480801307116772351524606, −6.93581265305444503586209067264, −5.40790710742987266672170214331, −4.97403477740751779734454742579, −4.56373623536873977020687715815, −3.15567976811670029058791400938, −2.13662955826695852375662986691, 0,
2.13662955826695852375662986691, 3.15567976811670029058791400938, 4.56373623536873977020687715815, 4.97403477740751779734454742579, 5.40790710742987266672170214331, 6.93581265305444503586209067264, 7.78654480801307116772351524606, 8.073291216211193933371132679457, 10.46621144258647288430662866189
