L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.292 + 0.707i)11-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (−0.292 − 0.707i)29-s + (−1.70 − 0.707i)37-s + (−0.707 + 1.70i)43-s + 1.00i·49-s + (0.707 − 1.70i)53-s − 1.00·63-s + (0.707 + 1.70i)67-s + (−0.707 + 0.292i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.292 + 0.707i)11-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (−0.292 − 0.707i)29-s + (−1.70 − 0.707i)37-s + (−0.707 + 1.70i)43-s + 1.00i·49-s + (0.707 − 1.70i)53-s − 1.00·63-s + (0.707 + 1.70i)67-s + (−0.707 + 0.292i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.033998110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033998110\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 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
−8.784149542408428064574056818915, −8.216022159784558551796076787300, −7.60554102131757339909422918228, −6.80312267654994533734741507548, −5.65278996164333649935611356501, −5.32818343898404828859655883619, −4.50302636536104243442070786687, −3.41710713307571425326477592164, −2.36918018122863424262731986994, −1.70751372482786433314108241827,
0.57781074874493026118869097139, 1.91762312295104800324687548662, 3.07791684451694665618487169571, 3.80511738776761967665269351208, 4.74231438930181858769852917685, 5.49851462577772948782722030414, 6.34760498661540420891405430479, 6.99690807360038076427341546263, 7.906605524689252486494547971968, 8.597206857010981855757307804448