L(s) = 1 | + (0.690 − 2.12i)2-s + (1.30 − 0.951i)3-s + (−2.42 − 1.76i)4-s + (−0.309 − 0.951i)5-s + (−1.11 − 3.44i)6-s + (0.809 + 0.587i)7-s + (−1.80 + 1.31i)8-s + (−0.118 + 0.363i)9-s − 2.23·10-s + (−2.54 − 2.12i)11-s − 4.85·12-s + (0.736 − 2.26i)13-s + (1.80 − 1.31i)14-s + (−1.30 − 0.951i)15-s + (−0.309 − 0.951i)16-s + (1.04 + 3.21i)17-s + ⋯ |
L(s) = 1 | + (0.488 − 1.50i)2-s + (0.755 − 0.549i)3-s + (−1.21 − 0.881i)4-s + (−0.138 − 0.425i)5-s + (−0.456 − 1.40i)6-s + (0.305 + 0.222i)7-s + (−0.639 + 0.464i)8-s + (−0.0393 + 0.121i)9-s − 0.707·10-s + (−0.767 − 0.641i)11-s − 1.40·12-s + (0.204 − 0.628i)13-s + (0.483 − 0.351i)14-s + (−0.337 − 0.245i)15-s + (−0.0772 − 0.237i)16-s + (0.253 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.298064 - 2.02578i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.298064 - 2.02578i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (2.54 + 2.12i)T \) |
good | 2 | \( 1 + (-0.690 + 2.12i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (-0.736 + 2.26i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 3.21i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1 - 0.726i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + (-4.73 - 3.44i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.381 - 1.17i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 1.62i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.23 + 3.80i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 + (-6.35 + 4.61i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.23 - 9.95i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.85 - 2.80i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.85 + 8.78i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + (-0.208 - 0.640i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.97 + 5.06i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.26 - 13.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.26 + 10.0i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + (-0.100 + 0.310i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.81871360819113563437085169509, −10.50548110551629010821743315251, −9.066074106837625225661995241798, −8.366881108472514915571235459804, −7.47435133098039371807705622699, −5.69451311749721211259173591532, −4.70839844352746932837507132591, −3.35631736301862010232394160492, −2.53459255338271896022783666608, −1.24784745694020690069363254571,
2.74775520243217931695804236082, 4.13496085109224146090348116090, 4.86112180311145715364746575055, 6.12917883158057356517935542873, 7.11475540424131652467184041204, 7.81487448181098685565717884188, 8.747886137935720709431429350992, 9.610641997105696863806906400518, 10.69587815479883717010490791087, 11.82617190786002253864772563105