L(s) = 1 | + (−1.17 + 0.780i)2-s + (1.51 + 0.848i)3-s + (0.780 − 1.84i)4-s + (−2.44 + 0.179i)6-s − 3.02i·7-s + (0.516 + 2.78i)8-s + (1.56 + 2.56i)9-s + 1.32·11-s + (2.74 − 2.11i)12-s + 5.12·13-s + (2.35 + 3.56i)14-s + (−2.78 − 2.87i)16-s + 2i·17-s + (−3.84 − 1.80i)18-s − 1.32i·19-s + ⋯ |
L(s) = 1 | + (−0.833 + 0.552i)2-s + (0.871 + 0.489i)3-s + (0.390 − 0.920i)4-s + (−0.997 + 0.0731i)6-s − 1.14i·7-s + (0.182 + 0.983i)8-s + (0.520 + 0.853i)9-s + 0.399·11-s + (0.791 − 0.611i)12-s + 1.42·13-s + (0.630 + 0.951i)14-s + (−0.695 − 0.718i)16-s + 0.485i·17-s + (−0.905 − 0.424i)18-s − 0.303i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17873 + 0.402476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17873 + 0.402476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.780i)T \) |
| 3 | \( 1 + (-1.51 - 0.848i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.02iT - 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 + 0.371T + 23T^{2} \) |
| 29 | \( 1 + 3.12iT - 29T^{2} \) |
| 31 | \( 1 - 4.71iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 - 7.73iT - 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 4.34iT - 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 8.10iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−11.34031506185000029683653353792, −10.60099224243916951649087567544, −9.888195531669791162901254543061, −8.864689523324308638447612517608, −8.185214050457426792952148419244, −7.21950962792513002388126894916, −6.21281630154379773597355293168, −4.64225323437421005847002337631, −3.45823220178323003835659713269, −1.50222307256617399489668037956,
1.53736606286210913618572713178, 2.79038990651337344931056864529, 3.88413711283470981255955176605, 5.97273721688449261049088032353, 7.05608149256728945203329408191, 8.182948738038439037982727865948, 8.852496794027586327674932395755, 9.425202070938424417768502399993, 10.66237697499104107972629122938, 11.74763724835736426913013200276