| L(s) = 1 | + 1.19·3-s − 1.56·5-s + 0.868i·7-s − 1.56·9-s − 3.09i·11-s − 4.74i·13-s − 1.87·15-s − 17-s + (−3.07 + 3.09i)19-s + 1.04i·21-s − 3.96i·23-s − 2.56·25-s − 5.47·27-s + 8.45i·29-s − 4.27·31-s + ⋯ |
| L(s) = 1 | + 0.692·3-s − 0.698·5-s + 0.328i·7-s − 0.520·9-s − 0.932i·11-s − 1.31i·13-s − 0.483·15-s − 0.242·17-s + (−0.704 + 0.709i)19-s + 0.227i·21-s − 0.825i·23-s − 0.512·25-s − 1.05·27-s + 1.57i·29-s − 0.767·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7449923057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7449923057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.07 - 3.09i)T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 0.868iT - 7T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 + 4.74iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 3.96iT - 23T^{2} \) |
| 29 | \( 1 - 8.45iT - 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 + 3.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.70iT - 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 + 9.27iT - 47T^{2} \) |
| 53 | \( 1 + 1.04iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.684T + 61T^{2} \) |
| 67 | \( 1 - 9.74T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 + 9.65iT - 83T^{2} \) |
| 89 | \( 1 + 5.79iT - 89T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
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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
−9.061647549898115873929987575577, −8.580288456391617876397782175322, −8.025979343351487476760388348632, −7.15437239091296632765566978379, −5.92335821168823465922958170914, −5.31287674166377855138736226415, −3.83877029815780430779955792941, −3.27996004811677026887283587861, −2.19666525930915372634763662125, −0.27093893868677562968284057942,
1.82170974936525142280637151218, 2.86408456586488144597702156272, 4.09335871624689699623579309510, 4.51784044909819092541119157363, 5.96806153308317868381592019565, 6.91767070574858640592721723215, 7.69091986526395459399244027326, 8.312380967080234817048354142500, 9.364724938224173507467157357440, 9.645171935370689604975524385699
