L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s − 4·11-s − 6i·13-s − 14-s + 16-s + 2i·17-s − 4i·22-s + 6·26-s − i·28-s + 6·29-s + 8·31-s + i·32-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.852i·22-s + 1.17·26-s − 0.188i·28-s + 1.11·29-s + 1.43·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219013087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219013087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.620458847782727851776238916580, −7.982779834373795867493677771238, −7.76089125495258490407748005774, −6.49108766081539479049912208082, −6.00102859239398283377855456955, −5.10620748871220915446797362532, −4.63154069072538103516153316650, −3.22581543751014398934573656441, −2.65848868802849023646703715401, −0.993112740136791518600613631740,
0.45150450724187887527511920428, 1.84329559571027258917234335505, 2.63676871248544139266347366121, 3.63618017843396477109989720116, 4.55820511930585603030608886214, 5.06092795556312604874733531608, 6.20501446670060598078154713008, 6.97695876416584513340373372504, 7.78895582283566919281588038133, 8.536377015452642172596434025106