L(s) = 1 | + (1.40 − 0.159i)2-s − 3-s + (1.94 − 0.449i)4-s + 0.947i·5-s + (−1.40 + 0.159i)6-s + 0.0251·7-s + (2.66 − 0.943i)8-s + 9-s + (0.151 + 1.33i)10-s + 1.77·11-s + (−1.94 + 0.449i)12-s − 5.89i·13-s + (0.0352 − 0.00401i)14-s − 0.947i·15-s + (3.59 − 1.75i)16-s + 2.08·17-s + ⋯ |
L(s) = 1 | + (0.993 − 0.113i)2-s − 0.577·3-s + (0.974 − 0.224i)4-s + 0.423i·5-s + (−0.573 + 0.0652i)6-s + 0.00949·7-s + (0.942 − 0.333i)8-s + 0.333·9-s + (0.0478 + 0.420i)10-s + 0.536·11-s + (−0.562 + 0.129i)12-s − 1.63i·13-s + (0.00943 − 0.00107i)14-s − 0.244i·15-s + (0.899 − 0.437i)16-s + 0.505·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.60425 - 0.252683i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60425 - 0.252683i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.159i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (7.47 + 3.33i)T \) |
good | 5 | \( 1 - 0.947iT - 5T^{2} \) |
| 7 | \( 1 - 0.0251T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 5.89iT - 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 4.96iT - 19T^{2} \) |
| 23 | \( 1 - 4.05iT - 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 - 5.82T + 37T^{2} \) |
| 41 | \( 1 - 6.60iT - 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 7.44iT - 53T^{2} \) |
| 59 | \( 1 + 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 2.42iT - 61T^{2} \) |
| 71 | \( 1 - 1.82iT - 71T^{2} \) |
| 73 | \( 1 + 9.08T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 9.99T + 89T^{2} \) |
| 97 | \( 1 + 3.04iT - 97T^{2} \) |
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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
−10.31406606143309409560048834653, −9.920388152094385006758211684802, −8.257027687803053515271548110460, −7.43536280356336710007780834137, −6.49134175933622020574158202758, −5.73091452907411997576561901389, −4.99952570517827333238417473764, −3.74648737514014434572456629692, −2.94184356685440328740494465036, −1.31411386668856942105846531027,
1.41209145249051837638126203132, 2.84224690903354188106104920676, 4.35431545117535832440125418937, 4.64790499742034403569990416625, 5.87915382832811950144521413886, 6.65671290627054659829716250761, 7.29298586905142790506246119801, 8.606009589870327638563051267354, 9.433633049182183304830339447061, 10.61602614663685506554397246386