L(s) = 1 | + (1.5 + 2.59i)3-s + (−2 − 1.73i)7-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)11-s − 2·13-s + (1.5 + 2.59i)17-s + (−2.5 + 4.33i)19-s + (1.5 − 7.79i)21-s + (−1.5 + 2.59i)23-s − 9·27-s − 6·29-s + (0.5 + 0.866i)31-s + (−1.5 + 2.59i)33-s + (−2.5 + 4.33i)37-s + (−3 − 5.19i)39-s + ⋯ |
L(s) = 1 | + (0.866 + 1.49i)3-s + (−0.755 − 0.654i)7-s + (−1 + 1.73i)9-s + (0.150 + 0.261i)11-s − 0.554·13-s + (0.363 + 0.630i)17-s + (−0.573 + 0.993i)19-s + (0.327 − 1.70i)21-s + (−0.312 + 0.541i)23-s − 1.73·27-s − 1.11·29-s + (0.0898 + 0.155i)31-s + (−0.261 + 0.452i)33-s + (−0.410 + 0.711i)37-s + (−0.480 − 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.314273617\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.314273617\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 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
−9.951557107883136220901478614631, −9.384173760551883886338485093543, −8.452417284738415844048676469056, −7.77954381182709700534669919593, −6.73143181364307256976399491170, −5.62109851521756496915958159966, −4.67334725040166923800437325202, −3.74285229862067030346367920367, −3.38982081492977341361608835567, −2.01271593918107366333146534594,
0.43985898638767071116077523231, 2.01147723082133800464218375439, 2.70338721237779144279009942853, 3.60525821180646879202693484188, 5.14011500336487292009151597330, 6.21638233515775495439081897598, 6.79667888119964279304563455634, 7.53918369702528869890902434783, 8.314955947288639081695278425182, 9.116127967281315344394440667656