L(s) = 1 | + (0.192 − 0.334i)2-s + (0.983 − 1.42i)3-s + (0.925 + 1.60i)4-s + (−0.286 − 0.603i)6-s + (1.17 + 2.36i)7-s + 1.48·8-s + (−1.06 − 2.80i)9-s + (−2.20 + 1.27i)11-s + (3.19 + 0.257i)12-s + 3.06·13-s + (1.01 + 0.0640i)14-s + (−1.56 + 2.71i)16-s + (5.59 − 3.23i)17-s + (−1.14 − 0.185i)18-s + (1.03 + 0.597i)19-s + ⋯ |
L(s) = 1 | + (0.136 − 0.236i)2-s + (0.567 − 0.823i)3-s + (0.462 + 0.801i)4-s + (−0.116 − 0.246i)6-s + (0.444 + 0.895i)7-s + 0.525·8-s + (−0.354 − 0.934i)9-s + (−0.663 + 0.383i)11-s + (0.922 + 0.0743i)12-s + 0.850·13-s + (0.272 + 0.0171i)14-s + (−0.391 + 0.677i)16-s + (1.35 − 0.783i)17-s + (−0.269 − 0.0436i)18-s + (0.237 + 0.137i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15760 - 0.227338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15760 - 0.227338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.983 + 1.42i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.17 - 2.36i)T \) |
good | 2 | \( 1 + (-0.192 + 0.334i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.20 - 1.27i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 + (-5.59 + 3.23i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.03 - 0.597i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.52 - 2.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.77iT - 29T^{2} \) |
| 31 | \( 1 + (5.95 - 3.43i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.07 - 1.77i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 5.46iT - 43T^{2} \) |
| 47 | \( 1 + (2.78 + 1.61i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.62 + 11.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.98 - 3.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.08 + 4.67i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.04 - 1.75i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.921iT - 71T^{2} \) |
| 73 | \( 1 + (0.148 + 0.256i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.14 + 7.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.11iT - 83T^{2} \) |
| 89 | \( 1 + (9.41 - 16.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−11.19968060776270229780070030820, −9.852531691786581298092086798508, −8.840101871864182211855591731704, −7.907664400808571850283326962123, −7.58135515583743138981588455839, −6.33828160096147305859439495228, −5.29318175759614676944341691977, −3.67097796639943696171993881873, −2.75012672877664786032607930903, −1.71491983708777816774014428603,
1.46039126370814782193244410126, 3.10426223594588739677816053834, 4.22079322376156782669131624098, 5.26850368506769282473756006847, 6.08301983181701254678478100674, 7.47667844030045916911906800810, 8.078678634804023433802971732627, 9.234981518114710468818618179507, 10.21570396165560305507474871600, 10.74590295087019853039054738067