L(s) = 1 | + i·2-s − 4-s − 2.29·5-s + 2.05·7-s − i·8-s − 2.29i·10-s − 0.573·11-s − 5.56i·13-s + 2.05i·14-s + 16-s + 5.74i·17-s + 4.17·19-s + 2.29·20-s − 0.573i·22-s − 2.06·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.02·5-s + 0.776·7-s − 0.353i·8-s − 0.726i·10-s − 0.173·11-s − 1.54i·13-s + 0.549i·14-s + 0.250·16-s + 1.39i·17-s + 0.958·19-s + 0.513·20-s − 0.122i·22-s − 0.429·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.001591530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001591530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 223 | \( 1 + (-2.79 - 14.6i)T \) |
good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 0.573T + 11T^{2} \) |
| 13 | \( 1 + 5.56iT - 13T^{2} \) |
| 17 | \( 1 - 5.74iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 2.06T + 23T^{2} \) |
| 29 | \( 1 - 3.14iT - 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 - 0.611T + 37T^{2} \) |
| 41 | \( 1 - 1.16iT - 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 6.02iT - 47T^{2} \) |
| 53 | \( 1 + 6.82iT - 53T^{2} \) |
| 59 | \( 1 - 1.99T + 59T^{2} \) |
| 61 | \( 1 + 14.1iT - 61T^{2} \) |
| 67 | \( 1 - 6.44iT - 67T^{2} \) |
| 71 | \( 1 - 0.0503T + 71T^{2} \) |
| 73 | \( 1 + 0.158T + 73T^{2} \) |
| 79 | \( 1 - 3.77iT - 79T^{2} \) |
| 83 | \( 1 + 3.37iT - 83T^{2} \) |
| 89 | \( 1 + 8.91iT - 89T^{2} \) |
| 97 | \( 1 + 9.51iT - 97T^{2} \) |
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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
−8.224457513496044451995685128550, −7.79462607024949926089249089653, −7.07319957708493075546294485255, −6.11176782223830988210462835327, −5.34692080511540219822340748352, −4.75043222833300682622069391885, −3.76151663822825968199477476902, −3.18660694362851223695082324377, −1.64569445355170666994893984933, −0.34683557368128789732309160012,
1.02081686501208716482273009312, 2.10488540697839189267683935192, 3.05994496790898176728360173138, 4.00667926232833788108834456869, 4.59045594800139258171673017228, 5.22091516802810377511406746913, 6.39462966744593465838102159955, 7.34582839144501527140751762284, 7.78773809569349513910142836062, 8.572130008286887552758131958773