L(s) = 1 | + 2.99i·3-s − 5-s + (0.183 − 2.63i)7-s − 5.98·9-s − 4.87·11-s + 2.42·13-s − 2.99i·15-s − 3.92i·17-s − 5.24i·19-s + (7.91 + 0.549i)21-s + 0.114i·23-s + 25-s − 8.94i·27-s − 3.60i·29-s + 4.62·31-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 0.447·5-s + (0.0693 − 0.997i)7-s − 1.99·9-s − 1.47·11-s + 0.673·13-s − 0.773i·15-s − 0.953i·17-s − 1.20i·19-s + (1.72 + 0.119i)21-s + 0.0237i·23-s + 0.200·25-s − 1.72i·27-s − 0.670i·29-s + 0.831·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6888289735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6888289735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.183 + 2.63i)T \) |
good | 3 | \( 1 - 2.99iT - 3T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 3.92iT - 17T^{2} \) |
| 19 | \( 1 + 5.24iT - 19T^{2} \) |
| 23 | \( 1 - 0.114iT - 23T^{2} \) |
| 29 | \( 1 + 3.60iT - 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 0.668iT - 53T^{2} \) |
| 59 | \( 1 + 1.37iT - 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + 9.43iT - 71T^{2} \) |
| 73 | \( 1 - 5.62iT - 73T^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 + 5.52iT - 83T^{2} \) |
| 89 | \( 1 + 6.21iT - 89T^{2} \) |
| 97 | \( 1 + 7.85iT - 97T^{2} \) |
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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
−9.920756667055171140264070826279, −9.051468095613543859549226883857, −8.164710806726777837368113166634, −7.41531441274846620235205843370, −6.18203597933096694740873722438, −4.86172648075735121987867540538, −4.71606107979493401805558744042, −3.55947761729626116901043521975, −2.80302685576005268259537575714, −0.30316440824889368243273170751,
1.44272426362794815449177917559, 2.41698675675498187091954260413, 3.42294010205379705261964057828, 5.12299794313963524190205061080, 5.91693768259718424337808919774, 6.59146653199721956713485182372, 7.61641548401717083262167515537, 8.370241677743183591985087432014, 8.463621567991365993093076518597, 10.03556596613592029131399806737