L(s) = 1 | − 2.38·5-s − i·7-s + 1.65i·11-s + 0.253i·13-s − 2i·17-s − 7.03·19-s + 2.82·23-s + 0.692·25-s + 1.26·29-s − 0.746i·31-s + 2.38i·35-s + 6.94i·37-s + 5.55i·41-s − 8.67·43-s + 10.9·47-s + ⋯ |
L(s) = 1 | − 1.06·5-s − 0.377i·7-s + 0.498i·11-s + 0.0702i·13-s − 0.485i·17-s − 1.61·19-s + 0.589·23-s + 0.138·25-s + 0.235·29-s − 0.134i·31-s + 0.403i·35-s + 1.14i·37-s + 0.867i·41-s − 1.32·43-s + 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.099030517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099030517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.38T + 5T^{2} \) |
| 11 | \( 1 - 1.65iT - 11T^{2} \) |
| 13 | \( 1 - 0.253iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 + 0.746iT - 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 5.55iT - 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 4.53iT - 61T^{2} \) |
| 67 | \( 1 + 0.253T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 + 14.5iT - 79T^{2} \) |
| 83 | \( 1 + 1.77iT - 83T^{2} \) |
| 89 | \( 1 + 3.13iT - 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−8.083679067137710054894274687669, −7.30178780824970163248349817780, −6.79888101644030791953918322030, −6.04022650653550618500942447924, −4.87680711370119957624999130869, −4.44468225215080083094621476762, −3.69779969072359597256443424543, −2.84854939083634840573830341900, −1.78649613979293731993243945337, −0.50314665263146546899600447676,
0.57624787825082764553617777649, 1.95590941314841440997262091269, 2.87099715177283997738541561912, 3.85184072308218260393798588702, 4.23078774854954371359444509129, 5.27644143235792880972701729441, 5.95882970220985249427372870292, 6.77891507493532385183967446923, 7.40124296740184832766202033517, 8.285750141447135330333890254364