L(s) = 1 | + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)25-s + 2·29-s + 2·41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)61-s + (−0.499 + 0.866i)81-s + (1 + 1.73i)89-s + (1 − 1.73i)101-s + (1 − 1.73i)109-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)25-s + 2·29-s + 2·41-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)61-s + (−0.499 + 0.866i)81-s + (1 + 1.73i)89-s + (1 − 1.73i)101-s + (1 − 1.73i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.243046349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243046349\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.476584581521341309333890046944, −7.955275465023248645659869494878, −7.32983875188193982980792942525, −6.42174566478572999440638066040, −5.51556473355296007245095231125, −4.68396227160948575842711348784, −4.31122266114823230233121323007, −3.17364617860842158216506235416, −2.08894897484033772793513713653, −0.976050005442044870313045094085,
0.990578571973092936812892467527, 2.45327454368982615807571498187, 3.21928571733310263959036936247, 4.06327471098678505270898339461, 4.71309788299406934760416163815, 6.00618644976148158386440807334, 6.44553819269419844744212034115, 7.24802992507568953803685822014, 7.77915737582202487182340114251, 8.702492116265349537293042649560