| 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
−8.285750141447135330333890254364, −7.40124296740184832766202033517, −6.77891507493532385183967446923, −5.95882970220985249427372870292, −5.27644143235792880972701729441, −4.23078774854954371359444509129, −3.85184072308218260393798588702, −2.87099715177283997738541561912, −1.95590941314841440997262091269, −0.57624787825082764553617777649,
0.50314665263146546899600447676, 1.78649613979293731993243945337, 2.84854939083634840573830341900, 3.69779969072359597256443424543, 4.44468225215080083094621476762, 4.87680711370119957624999130869, 6.04022650653550618500942447924, 6.79888101644030791953918322030, 7.30178780824970163248349817780, 8.083679067137710054894274687669
