L(s) = 1 | + 2.21·5-s − i·7-s − 2.42i·11-s + 1.11i·13-s + 0.701i·17-s − 0.938·19-s − 4.30·23-s − 0.102·25-s − 4.26·29-s − 7.02i·31-s − 2.21i·35-s − 3.12i·37-s + 0.157i·41-s − 7.08·43-s + 0.867·47-s + ⋯ |
L(s) = 1 | + 0.989·5-s − 0.377i·7-s − 0.730i·11-s + 0.309i·13-s + 0.170i·17-s − 0.215·19-s − 0.897·23-s − 0.0204·25-s − 0.791·29-s − 1.26i·31-s − 0.374i·35-s − 0.513i·37-s + 0.0246i·41-s − 1.08·43-s + 0.126·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.152514397\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152514397\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 + 2.42iT - 11T^{2} \) |
| 13 | \( 1 - 1.11iT - 13T^{2} \) |
| 17 | \( 1 - 0.701iT - 17T^{2} \) |
| 19 | \( 1 + 0.938T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.02iT - 31T^{2} \) |
| 37 | \( 1 + 3.12iT - 37T^{2} \) |
| 41 | \( 1 - 0.157iT - 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 - 0.867T + 47T^{2} \) |
| 53 | \( 1 + 3.91T + 53T^{2} \) |
| 59 | \( 1 + 7.28iT - 59T^{2} \) |
| 61 | \( 1 + 4.57iT - 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 - 6.65iT - 79T^{2} \) |
| 83 | \( 1 + 8.41iT - 83T^{2} \) |
| 89 | \( 1 + 7.34iT - 89T^{2} \) |
| 97 | \( 1 - 7.99T + 97T^{2} \) |
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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
−7.86138620748581655292298135350, −7.05810529644761388844770796878, −6.14048838907830694891002363838, −5.91280312419149495977808627288, −4.99204083241242242046264261603, −4.08765827413360421666438170466, −3.37350586102892735928081303449, −2.26226368734462327770193587899, −1.61621265876316809580060611577, −0.26011054084795520428102990914,
1.44163977282570425512412250832, 2.12723393040136964361158182728, 2.97604807867144771524684439824, 3.96900172404923459133670294158, 4.90578166015155402402183502333, 5.48855414784291594728759683501, 6.19686622252885222117897090747, 6.82258763823484395706113826895, 7.65186586828742041711426316035, 8.351502050128772051027558450906