L(s) = 1 | + (2.87 − 1.65i)3-s + (2.49 + 0.873i)7-s + (3.99 − 6.92i)9-s + (2.12 + 3.67i)11-s − 5.10i·13-s + (−0.765 + 0.441i)17-s + (−3.27 + 5.66i)19-s + (8.61 − 1.63i)21-s + (4.62 + 2.66i)23-s − 16.5i·27-s − 4.54·29-s + (−0.338 − 0.586i)31-s + (12.1 + 7.03i)33-s + (−1.31 − 0.760i)37-s + (−8.46 − 14.6i)39-s + ⋯ |
L(s) = 1 | + (1.65 − 0.957i)3-s + (0.943 + 0.330i)7-s + (1.33 − 2.30i)9-s + (0.640 + 1.10i)11-s − 1.41i·13-s + (−0.185 + 0.107i)17-s + (−0.750 + 1.29i)19-s + (1.88 − 0.355i)21-s + (0.963 + 0.556i)23-s − 3.18i·27-s − 0.843·29-s + (−0.0608 − 0.105i)31-s + (2.12 + 1.22i)33-s + (−0.216 − 0.125i)37-s + (−1.35 − 2.34i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.408564575\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.408564575\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.49 - 0.873i)T \) |
good | 3 | \( 1 + (-2.87 + 1.65i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.10iT - 13T^{2} \) |
| 17 | \( 1 + (0.765 - 0.441i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.27 - 5.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.62 - 2.66i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 + (0.338 + 0.586i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.31 + 0.760i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 - 5.49iT - 43T^{2} \) |
| 47 | \( 1 + (6.29 + 3.63i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.22 - 1.28i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.97 + 6.88i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.50 + 7.81i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.63 + 1.52i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 + (3.34 - 1.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.352 + 0.610i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + (4.20 - 7.27i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.614iT - 97T^{2} \) |
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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
−9.325827385467436554908216554633, −8.326484775343471121739906614681, −8.052355998470782647272842353798, −7.27976514775612896074874371296, −6.47313781863460860648842214351, −5.24971695519653210124972111871, −4.03958957337904131629904697502, −3.18987777976172716021021655351, −2.05448974301578234941116427715, −1.43596008550232007000982910873,
1.64545079036945265016236487549, 2.65324613427782170720571094921, 3.70101099086600868017490829129, 4.39029868119294930735683638504, 5.06101528946935167315702853927, 6.66505206272351404113656209064, 7.41428809517831450671304314383, 8.494567785763201383976039292422, 8.819562620674740145622877353138, 9.321336937394154043525459501086