| L(s) = 1 | − 4.89·2-s − 6.57·3-s + 15.9·4-s − 15.5·5-s + 32.1·6-s − 38.7·8-s + 16.1·9-s + 76.0·10-s − 11·11-s − 104.·12-s − 74.3·13-s + 102.·15-s + 62.1·16-s + 94.0·17-s − 79.2·18-s − 135.·19-s − 247.·20-s + 53.8·22-s + 81.1·23-s + 254.·24-s + 116.·25-s + 363.·26-s + 70.9·27-s − 53.4·29-s − 499.·30-s + 9.50·31-s + 6.09·32-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 1.26·3-s + 1.99·4-s − 1.39·5-s + 2.18·6-s − 1.71·8-s + 0.599·9-s + 2.40·10-s − 0.301·11-s − 2.51·12-s − 1.58·13-s + 1.75·15-s + 0.970·16-s + 1.34·17-s − 1.03·18-s − 1.63·19-s − 2.76·20-s + 0.521·22-s + 0.735·23-s + 2.16·24-s + 0.934·25-s + 2.74·26-s + 0.506·27-s − 0.342·29-s − 3.04·30-s + 0.0550·31-s + 0.0336·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 3 | \( 1 + 6.57T + 27T^{2} \) |
| 5 | \( 1 + 15.5T + 125T^{2} \) |
| 13 | \( 1 + 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.50T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.14T + 5.06e4T^{2} \) |
| 41 | \( 1 - 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 694.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 278.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 886.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 185.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 847.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−10.13740992199089157446414531882, −9.105234503922666264172200738125, −8.051521340674134442543033372534, −7.48077104042567469591784621353, −6.73100608269957684695051604806, −5.49773603614173117459254845016, −4.30551314122724776551806868186, −2.58049135459312609771529425922, −0.78620168823730262012030385833, 0,
0.78620168823730262012030385833, 2.58049135459312609771529425922, 4.30551314122724776551806868186, 5.49773603614173117459254845016, 6.73100608269957684695051604806, 7.48077104042567469591784621353, 8.051521340674134442543033372534, 9.105234503922666264172200738125, 10.13740992199089157446414531882
