| L(s) = 1 | + i·5-s + (−2.44 − i)7-s − 2.44i·11-s + 4.44i·13-s − 6.89i·17-s + 4.44·19-s + 2i·23-s − 25-s + 6.89·29-s − 8.44·31-s + (1 − 2.44i)35-s + 6.89·37-s − 0.898i·41-s + 8i·43-s − 10.8·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + (−0.925 − 0.377i)7-s − 0.738i·11-s + 1.23i·13-s − 1.67i·17-s + 1.02·19-s + 0.417i·23-s − 0.200·25-s + 1.28·29-s − 1.51·31-s + (0.169 − 0.414i)35-s + 1.13·37-s − 0.140i·41-s + 1.21i·43-s − 1.58·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8617163830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8617163830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.44 + i)T \) |
| good | 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 6.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.44T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 + 0.898iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 12.8iT - 61T^{2} \) |
| 67 | \( 1 + 15.7iT - 67T^{2} \) |
| 71 | \( 1 + 6.44iT - 71T^{2} \) |
| 73 | \( 1 - 5.34iT - 73T^{2} \) |
| 79 | \( 1 + 9.79iT - 79T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 + 15.7iT - 89T^{2} \) |
| 97 | \( 1 + 2.65iT - 97T^{2} \) |
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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
−7.79069080496879827299031579835, −7.27862995282422901853912474652, −6.56966226774774897768807993415, −6.04556646249398503124329911588, −5.03716508704545413344171983321, −4.27991440720363098995316670129, −3.17476686192139266355537929114, −2.94205284284859347523633900128, −1.51227194499763577830769106397, −0.25215836141642720255410824053,
1.10509794695140586950184648073, 2.23988732184253776654586199273, 3.20906622103601452800787661221, 3.86730803271881813378660316222, 4.90550733485981515304289667271, 5.57075626307488376934795879263, 6.25930346830038984124070584949, 6.95475452410035741574159458674, 7.949721875606733321024512225122, 8.294318859091213755903408953118
