L(s) = 1 | + 2-s − 4-s − 1.41i·7-s − 3·8-s + (3 − 1.41i)11-s + 2.82i·13-s − 1.41i·14-s − 16-s + 2·17-s − 4.24i·19-s + (3 − 1.41i)22-s + 2.82i·26-s + 1.41i·28-s − 6·29-s + 5·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.5·4-s − 0.534i·7-s − 1.06·8-s + (0.904 − 0.426i)11-s + 0.784i·13-s − 0.377i·14-s − 0.250·16-s + 0.485·17-s − 0.973i·19-s + (0.639 − 0.301i)22-s + 0.554i·26-s + 0.267i·28-s − 1.11·29-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.532598872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.532598872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-3 + 1.41i)T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 12.7iT - 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.947429024077156820938337818459, −7.973457791729002794039568640565, −6.93781539653073318982600137940, −6.40950927484118579323489048208, −5.39146171517367558249847039900, −4.73872779296264986644860584122, −3.79573072757325229596260418073, −3.37323220293986808110865779505, −1.87650461439571599527840217535, −0.43328419087529532887566205777,
1.32763939073741645243102507143, 2.70317280267051485689814266796, 3.65043888187216562916594952017, 4.25223124601361947721854785288, 5.39788074511568265068825374384, 5.69690560716048593252894151112, 6.65693434760224074119764483491, 7.63862135316422984781261979684, 8.461342399726571342192181737833, 9.128653208233236826055818918593