L(s) = 1 | − 0.131i·2-s + 1.44i·3-s + 1.98·4-s + (−2.22 − 0.230i)5-s + 0.189·6-s − 0.456i·7-s − 0.524i·8-s + 0.925·9-s + (−0.0303 + 0.292i)10-s + 11-s + 2.85i·12-s − 5.24i·13-s − 0.0601·14-s + (0.331 − 3.20i)15-s + 3.89·16-s − 6.95i·17-s + ⋯ |
L(s) = 1 | − 0.0930i·2-s + 0.831i·3-s + 0.991·4-s + (−0.994 − 0.103i)5-s + 0.0773·6-s − 0.172i·7-s − 0.185i·8-s + 0.308·9-s + (−0.00958 + 0.0925i)10-s + 0.301·11-s + 0.824i·12-s − 1.45i·13-s − 0.0160·14-s + (0.0857 − 0.827i)15-s + 0.974·16-s − 1.68i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.847193889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847193889\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.22 + 0.230i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 0.131iT - 2T^{2} \) |
| 3 | \( 1 - 1.44iT - 3T^{2} \) |
| 7 | \( 1 + 0.456iT - 7T^{2} \) |
| 13 | \( 1 + 5.24iT - 13T^{2} \) |
| 17 | \( 1 + 6.95iT - 17T^{2} \) |
| 23 | \( 1 - 3.13iT - 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 + 0.922iT - 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 0.870iT - 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 8.75iT - 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 1.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 + 2.75iT - 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 - 4.84iT - 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 9.50iT - 97T^{2} \) |
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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
−10.04018833194300805622735733425, −9.268515099740047496327974899202, −8.054521749977251308584327772892, −7.46405247996009707593226359259, −6.71459933760684991233041353153, −5.42551899459171290632562298265, −4.61729359730364599595210369858, −3.50155229196107191730881426161, −2.86593118262433756615389166489, −0.964290341612790503501445626874,
1.33733920486374301276467875477, 2.30852590378305600227902621241, 3.65512019260960383040604090726, 4.54080846976218467180262740559, 6.13570067311835092299681748512, 6.67071590982194149182856699473, 7.28714360025517018988760111738, 8.120839844334275820871485995735, 8.785420806725388005079129999571, 10.16075972293688939463807561789