L(s) = 1 | − 2.82i·5-s + 1.41i·7-s + 2·13-s − 1.41i·17-s + 2.82i·19-s − 6·23-s − 3.00·25-s + 5.65i·29-s + 9.89i·31-s + 4.00·35-s − 2·37-s + 4.24i·41-s + 8.48i·43-s − 10·47-s + 5·49-s + ⋯ |
L(s) = 1 | − 1.26i·5-s + 0.534i·7-s + 0.554·13-s − 0.342i·17-s + 0.648i·19-s − 1.25·23-s − 0.600·25-s + 1.05i·29-s + 1.77i·31-s + 0.676·35-s − 0.328·37-s + 0.662i·41-s + 1.29i·43-s − 1.45·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.440204214\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.440204214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.89iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 12.7iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.514279705462625924714808563202, −7.973585789856338936828036925021, −6.93192915184719534918215991765, −6.15975675213146004532948389346, −5.34930459758865093167425798010, −4.88717081131835176986691911032, −3.95453226816894290730954599198, −3.10480788238594224027819840609, −1.86607018937731840724692738260, −1.08652884750568271833373000645,
0.43000508201211822424306888142, 1.96826483377392857615872328186, 2.70818312196148287393452785584, 3.79972161800046962695423851304, 4.11834416624773577878285300205, 5.43425506182866440690025483124, 6.16362705844462686329400091128, 6.75383540289388016043689569726, 7.44099731351353017055810124025, 8.063185903273047506396163996512