L(s) = 1 | + (1.04 − 1.80i)2-s + (−1.04 + 1.38i)3-s + (−1.17 − 2.03i)4-s + (−0.5 − 0.866i)5-s + (1.41 + 3.32i)6-s + (−2.04 + 3.53i)7-s − 0.734·8-s + (−0.824 − 2.88i)9-s − 2.08·10-s + (0.675 − 1.17i)11-s + (4.04 + 0.498i)12-s + (−0.324 − 0.561i)13-s + (4.26 + 7.38i)14-s + (1.71 + 0.211i)15-s + (1.58 − 2.74i)16-s − 1.35·17-s + ⋯ |
L(s) = 1 | + (0.737 − 1.27i)2-s + (−0.602 + 0.798i)3-s + (−0.587 − 1.01i)4-s + (−0.223 − 0.387i)5-s + (0.575 + 1.35i)6-s + (−0.772 + 1.33i)7-s − 0.259·8-s + (−0.274 − 0.961i)9-s − 0.659·10-s + (0.203 − 0.353i)11-s + (1.16 + 0.143i)12-s + (−0.0898 − 0.155i)13-s + (1.13 + 1.97i)14-s + (0.443 + 0.0547i)15-s + (0.396 − 0.686i)16-s − 0.327·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.802763 - 0.409533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.802763 - 0.409533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.04 - 1.38i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.04 + 1.80i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (2.04 - 3.53i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.675 + 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.324 + 0.561i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 + (2.39 + 4.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.93 - 3.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.84 - 6.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + (-0.0898 - 0.155i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.410 + 0.710i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.45 - 9.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.07 + 7.05i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + (5.17 - 8.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.12 + 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (-6.79 + 11.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.72480935386663217562861737340, −14.47351145043800891660113714211, −12.84287240466090350039328936299, −12.15809529035548401330996626524, −11.25263784495139753445860362544, −10.00807396555845522052903944309, −8.884947726371900626927213454153, −6.00939549294963706675553880929, −4.67761242782936460247511586809, −3.09312176866925448606064717414,
4.21214934995389426819785277996, 6.01865674004262494123430804984, 7.00572775022621672484051852226, 7.74022067356809975746706903007, 10.16218166475136889031121479286, 11.60867463198518106702045057164, 13.18616213902954527432610978770, 13.61889279812603858364531905490, 14.88630021574237651119322952539, 16.16411806774599775847909098682