| L(s) = 1 | − 1.22·5-s + (−1.48 − 2.19i)7-s + 3.31i·11-s − 1.60i·13-s + 6.11·17-s + 1.35i·19-s + 1.11i·23-s − 3.50·25-s − 9.39i·29-s − 7.00i·31-s + (1.81 + 2.67i)35-s − 11.7·37-s − 5.86·41-s − 8.58·43-s − 3.15·47-s + ⋯ |
| L(s) = 1 | − 0.546·5-s + (−0.560 − 0.828i)7-s + 0.998i·11-s − 0.443i·13-s + 1.48·17-s + 0.310i·19-s + 0.232i·23-s − 0.701·25-s − 1.74i·29-s − 1.25i·31-s + (0.306 + 0.452i)35-s − 1.92·37-s − 0.916·41-s − 1.30·43-s − 0.460·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5303816442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5303816442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.48 + 2.19i)T \) |
| good | 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 - 3.31iT - 11T^{2} \) |
| 13 | \( 1 + 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 - 1.35iT - 19T^{2} \) |
| 23 | \( 1 - 1.11iT - 23T^{2} \) |
| 29 | \( 1 + 9.39iT - 29T^{2} \) |
| 31 | \( 1 + 7.00iT - 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 5.86T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 + 0.539iT - 53T^{2} \) |
| 59 | \( 1 + 6.87T + 59T^{2} \) |
| 61 | \( 1 + 7.40iT - 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 - 4.81iT - 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 - 1.15T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 4.04iT - 97T^{2} \) |
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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
−9.361559846552185253131411341112, −7.944473085951685481580946976365, −7.76262263048998211938301941721, −6.82445332512620574091347909899, −5.91248991528861430187128762559, −4.88770600264428930614628599046, −3.89851226113674890036032177890, −3.27181612679264720725063037535, −1.76289108805430386505222867738, −0.20915700890585031198954082265,
1.54163190301626552776091796826, 3.16206609505434030640361476455, 3.47467820275076565314584765012, 4.98145800738976069318848018258, 5.62765882813959551848014094363, 6.59873758015174351840593778096, 7.32801743722409600849910991492, 8.539428139539736455533007377147, 8.689517891595165699145666189429, 9.806606307057680940808529261300
