L(s) = 1 | + (−0.866 + 1.5i)3-s + (−0.866 + 0.5i)5-s + (−1 − 1.73i)9-s + (−1.5 − 0.866i)11-s − i·13-s − 1.73i·15-s + (0.866 + 0.5i)17-s + (0.499 − 0.866i)25-s + 1.73·27-s + 29-s + (2.59 − 1.5i)33-s + (1.5 + 0.866i)39-s + (1.73 + i)45-s + (0.866 + 1.5i)47-s + (−1.5 + 0.866i)51-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.5i)3-s + (−0.866 + 0.5i)5-s + (−1 − 1.73i)9-s + (−1.5 − 0.866i)11-s − i·13-s − 1.73i·15-s + (0.866 + 0.5i)17-s + (0.499 − 0.866i)25-s + 1.73·27-s + 29-s + (2.59 − 1.5i)33-s + (1.5 + 0.866i)39-s + (1.73 + i)45-s + (0.866 + 1.5i)47-s + (−1.5 + 0.866i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5809612273\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5809612273\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−8.639544783168998539374511072890, −8.065516502128069817519554771972, −7.39470543249850889898632020134, −6.12847634956544679466214847759, −5.70487914526630575482755047929, −4.93492800529279464181311647437, −4.24782872207413838987047141355, −3.24575881232710713749357959971, −2.94748474161607105176198663380, −0.54202867160820004478156585201,
0.805452926310872466956013981577, 1.90983710187228794544647101830, 2.80808094018709957033736870909, 4.17421640244165999271667986347, 5.09249014686402229310431398467, 5.47027143955413791704570723321, 6.61769930245408829271391891347, 7.12972068286188832950809860991, 7.73956487255361889172426592037, 8.181006595244521038112189770501