L(s) = 1 | + (−1.38 − 0.303i)2-s − 1.34i·3-s + (1.81 + 0.837i)4-s + 5-s + (−0.406 + 1.85i)6-s + (−1.28 − 2.31i)7-s + (−2.25 − 1.70i)8-s + 1.20·9-s + (−1.38 − 0.303i)10-s + 2.44·11-s + (1.12 − 2.43i)12-s − 1.57·13-s + (1.06 + 3.58i)14-s − 1.34i·15-s + (2.59 + 3.04i)16-s + 1.11i·17-s + ⋯ |
L(s) = 1 | + (−0.976 − 0.214i)2-s − 0.774i·3-s + (0.908 + 0.418i)4-s + 0.447·5-s + (−0.166 + 0.756i)6-s + (−0.483 − 0.875i)7-s + (−0.797 − 0.603i)8-s + 0.400·9-s + (−0.436 − 0.0958i)10-s + 0.738·11-s + (0.324 − 0.703i)12-s − 0.435·13-s + (0.284 + 0.958i)14-s − 0.346i·15-s + (0.649 + 0.760i)16-s + 0.271i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.569583 - 0.657603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.569583 - 0.657603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 + 0.303i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.28 + 2.31i)T \) |
good | 3 | \( 1 + 1.34iT - 3T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 1.11iT - 17T^{2} \) |
| 19 | \( 1 + 8.44iT - 19T^{2} \) |
| 23 | \( 1 - 2.62iT - 23T^{2} \) |
| 29 | \( 1 + 3.43iT - 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 6.22iT - 37T^{2} \) |
| 41 | \( 1 - 3.13iT - 41T^{2} \) |
| 43 | \( 1 - 7.45T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 - 6.16iT - 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 + 2.15T + 67T^{2} \) |
| 71 | \( 1 - 7.87iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 7.70iT - 79T^{2} \) |
| 83 | \( 1 - 0.813iT - 83T^{2} \) |
| 89 | \( 1 - 5.12iT - 89T^{2} \) |
| 97 | \( 1 + 0.833iT - 97T^{2} \) |
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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
−11.43858983567817227103145982454, −10.59856449152034546376727004172, −9.591073892701813253505903048553, −8.958537305499851448459886028737, −7.33003990833043051680772368879, −7.18345737837795445918899795206, −6.03386043667015101428790968242, −4.03956189559516934302546570895, −2.40234878092203970455099851567, −0.954027433515987197822461561712,
1.91233050535756757058563382852, 3.55372481424725496181667774466, 5.27716517643229262787136656236, 6.21786107808791704218221489415, 7.29519388739031493769466712276, 8.618675618969095812454077153849, 9.395753246441666162513063117294, 9.970736857857461941879881645688, 10.79970708500516692860007082435, 12.01899212259584449391584649810