| L(s) = 1 | − 0.859i·2-s − i·3-s + 1.26·4-s − i·5-s − 0.859·6-s + (2.42 − 1.05i)7-s − 2.80i·8-s − 9-s − 0.859·10-s + (−3.22 − 0.765i)11-s − 1.26i·12-s − 3.49·13-s + (−0.904 − 2.08i)14-s − 15-s + 0.110·16-s − 5.22·17-s + ⋯ |
| L(s) = 1 | − 0.608i·2-s − 0.577i·3-s + 0.630·4-s − 0.447i·5-s − 0.351·6-s + (0.917 − 0.397i)7-s − 0.991i·8-s − 0.333·9-s − 0.271·10-s + (−0.972 − 0.230i)11-s − 0.363i·12-s − 0.968·13-s + (−0.241 − 0.557i)14-s − 0.258·15-s + 0.0275·16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593829667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593829667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.42 + 1.05i)T \) |
| 11 | \( 1 + (3.22 + 0.765i)T \) |
| good | 2 | \( 1 + 0.859iT - 2T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 6.88iT - 29T^{2} \) |
| 31 | \( 1 - 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 - 4.10iT - 43T^{2} \) |
| 47 | \( 1 + 6.68iT - 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 + 4.22iT - 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 + 6.63iT - 79T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 - 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−9.526947277581707272378369507563, −8.438259747814976419684501828009, −7.77216269034020031865851056667, −7.02428424455987529981175739362, −6.12939321848508429847226253734, −4.97869787063447705862332875397, −4.15497700318421503668671161887, −2.59427638470024116340254628217, −2.04846262788999292315728428717, −0.62462313808575463568739330442,
2.17911592180620439502154256025, 2.73342653998015654439309368106, 4.42741984222457463414860048241, 5.09572602976468873106630037795, 5.96871158223061870322150839239, 6.94650527622440387798299221188, 7.63313848757166835929579200820, 8.429820890383758948585014590809, 9.214301609924312110562939881491, 10.41932264170493112064619111585
