L(s) = 1 | + (−0.835 − 1.14i)2-s + (1.10 + 1.33i)3-s + (−0.602 + 1.90i)4-s + (−1 + 2i)5-s + (0.602 − 2.37i)6-s + 7-s + (2.67 − 0.907i)8-s + (−0.569 + 2.94i)9-s + (3.11 − 0.531i)10-s − 2.20·11-s + (−3.21 + 1.29i)12-s − 1.89i·13-s + (−0.835 − 1.14i)14-s + (−3.77 + 0.868i)15-s + (−3.27 − 2.29i)16-s − 1.13·17-s + ⋯ |
L(s) = 1 | + (−0.591 − 0.806i)2-s + (0.636 + 0.771i)3-s + (−0.301 + 0.953i)4-s + (−0.447 + 0.894i)5-s + (0.245 − 0.969i)6-s + 0.377·7-s + (0.947 − 0.320i)8-s + (−0.189 + 0.981i)9-s + (0.985 − 0.167i)10-s − 0.664·11-s + (−0.927 + 0.374i)12-s − 0.524i·13-s + (−0.223 − 0.304i)14-s + (−0.974 + 0.224i)15-s + (−0.818 − 0.574i)16-s − 0.276·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.711817 + 0.657239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.711817 + 0.657239i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.835 + 1.14i)T \) |
| 3 | \( 1 + (-1.10 - 1.33i)T \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 1.89iT - 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 8.56iT - 19T^{2} \) |
| 23 | \( 1 - 3.21iT - 23T^{2} \) |
| 29 | \( 1 + 1.89iT - 29T^{2} \) |
| 31 | \( 1 - 5.90iT - 31T^{2} \) |
| 37 | \( 1 - 0.409iT - 37T^{2} \) |
| 41 | \( 1 + 4.40iT - 41T^{2} \) |
| 43 | \( 1 + 0.934T + 43T^{2} \) |
| 47 | \( 1 + 2.67iT - 47T^{2} \) |
| 53 | \( 1 - 6.81T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 3.76iT - 79T^{2} \) |
| 83 | \( 1 - 6.84iT - 83T^{2} \) |
| 89 | \( 1 + 16.1iT - 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−11.16229498990974906212037669477, −10.23816234232850951173190229175, −10.07816970500568777753968936539, −8.636408556945828718324266004060, −8.031103993830072472734725825474, −7.25248724513174401802230219596, −5.43266642212026845225629409571, −4.03466984420156668370720869976, −3.28961342638728631178936007248, −2.14544638349669842420452634714,
0.70420965804479226476391564329, 2.29004087071605131451401318328, 4.30990576236128458921911936405, 5.28767148324572590132701262728, 6.61831166983850639860601132638, 7.41147692070654039884656663514, 8.264108872191281803347904761646, 8.855687301522025301282237298304, 9.601247310450138489351168565277, 11.00600383342732366283204271423