| L(s) = 1 | + i·3-s + 1.38i·7-s − 9-s + 6.23·11-s + 3i·13-s − 3.38i·17-s − 19-s − 1.38·21-s + 6.70i·23-s − i·27-s + 4.23·29-s + 3.09·31-s + 6.23i·33-s − 5i·37-s − 3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.522i·7-s − 0.333·9-s + 1.88·11-s + 0.832i·13-s − 0.820i·17-s − 0.229·19-s − 0.301·21-s + 1.39i·23-s − 0.192i·27-s + 0.786·29-s + 0.555·31-s + 1.08i·33-s − 0.821i·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.194872107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194872107\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.38iT - 7T^{2} \) |
| 11 | \( 1 - 6.23T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 3.38iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6.70iT - 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 2.38iT - 43T^{2} \) |
| 47 | \( 1 - 9.18iT - 47T^{2} \) |
| 53 | \( 1 + 3.61iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 - 9.14iT - 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 8.32iT - 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 9.56iT - 97T^{2} \) |
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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
−8.503319696765131294705433582317, −7.47974732061360613195214676689, −6.73908267041762519781100356696, −6.18130475766725826402720921661, −5.36166389742219699620843249776, −4.52404177533123052086773549903, −3.92770556282587765399507964437, −3.14516942269596473678648841214, −2.08903039400334965441480885432, −1.09247224217868012486904249346,
0.66049535098858556224487652619, 1.43505483538762821959349875866, 2.48481502600940957506789644544, 3.52634199587743765933659664259, 4.13977230156081707623358722189, 4.98242306379367774270142354107, 6.06864532716992106797738150033, 6.51979606830412014128565081458, 7.03724175706636410580252218956, 7.952537303851671843929230123238
