| L(s) = 1 | + i·3-s − 9-s + 4·11-s + 2i·13-s + 2i·17-s − 8·19-s + 4i·23-s − i·27-s + 6·29-s + 4i·33-s − 2i·37-s − 2·39-s − 6·41-s + 4i·43-s + 12i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 0.485i·17-s − 1.83·19-s + 0.834i·23-s − 0.192i·27-s + 1.11·29-s + 0.696i·33-s − 0.328i·37-s − 0.320·39-s − 0.937·41-s + 0.609i·43-s + 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422015497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422015497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.121279321727361796187495882666, −8.653946391450262269850011610284, −7.80110789286529074190981222881, −6.59253762829136573373045722505, −6.33179208332781865686470221518, −5.17977672209596164305187515504, −4.22025835593867010029583925639, −3.80609050352547795382138013611, −2.52612844324430918158692015706, −1.37731319719758585072401552200,
0.49200684404884067164167360943, 1.76875518447980779067063967005, 2.74778045478718415433800835631, 3.85858199368142659427749753835, 4.68200779026234768406191851070, 5.72318174511009985932257059866, 6.66034590661104915103867788837, 6.86163984448130820996516660373, 8.132350852563471362107334593035, 8.566199376355159919024556049375
