L(s) = 1 | + 1.83·2-s − 0.556i·3-s + 1.36·4-s + 5-s − 1.02i·6-s − 4.08i·7-s − 1.15·8-s + 2.69·9-s + 1.83·10-s + (2.99 − 1.43i)11-s − 0.760i·12-s − 5.36·13-s − 7.50i·14-s − 0.556i·15-s − 4.86·16-s − 5.36i·17-s + ⋯ |
L(s) = 1 | + 1.29·2-s − 0.321i·3-s + 0.684·4-s + 0.447·5-s − 0.416i·6-s − 1.54i·7-s − 0.409·8-s + 0.896·9-s + 0.580·10-s + (0.901 − 0.432i)11-s − 0.219i·12-s − 1.48·13-s − 2.00i·14-s − 0.143i·15-s − 1.21·16-s − 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.184670200\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.184670200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (-2.99 + 1.43i)T \) |
| 19 | \( 1 + (0.797 - 4.28i)T \) |
good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.556iT - 3T^{2} \) |
| 7 | \( 1 + 4.08iT - 7T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 + 5.36iT - 17T^{2} \) |
| 23 | \( 1 + 1.90T + 23T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 - 5.60iT - 31T^{2} \) |
| 37 | \( 1 - 3.42iT - 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 + 6.62iT - 43T^{2} \) |
| 47 | \( 1 - 7.01T + 47T^{2} \) |
| 53 | \( 1 + 7.74iT - 53T^{2} \) |
| 59 | \( 1 - 3.01iT - 59T^{2} \) |
| 61 | \( 1 - 7.25iT - 61T^{2} \) |
| 67 | \( 1 + 6.53iT - 67T^{2} \) |
| 71 | \( 1 + 0.908iT - 71T^{2} \) |
| 73 | \( 1 - 9.46iT - 73T^{2} \) |
| 79 | \( 1 + 9.42T + 79T^{2} \) |
| 83 | \( 1 - 3.44iT - 83T^{2} \) |
| 89 | \( 1 + 6.38iT - 89T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−9.969088742881435254090250446739, −9.062403358680827952963027950676, −7.67594890935869569036531554552, −6.96053605377955499861919318060, −6.42795959875392206234249678169, −5.18601356146120344470755713464, −4.43186455607918259834295895701, −3.75383025853626412823851588147, −2.52877374662886804701450093420, −0.988887527019411439073644520319,
2.05124154991885284880180029760, 2.82945849802479115484296964188, 4.27641911603150461731397347419, 4.66878286035708631776976765127, 5.74488585098025413438961705452, 6.30324995365796208812931436056, 7.29008880444080727037804828332, 8.656263584597300126263627552288, 9.387579104566490680971398065726, 9.912637172448787910435604371569