| L(s) = 1 | + (−1.48 − 1.67i)5-s + i·7-s + 6.31·11-s − 6.96i·13-s − 6.57i·17-s − 3.73·19-s − 5.73i·23-s + (−0.612 + 4.96i)25-s − 2·29-s + 1.03·31-s + (1.67 − 1.48i)35-s + 10.7i·37-s + 6.96·41-s − 5.92i·43-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.662 − 0.749i)5-s + 0.377i·7-s + 1.90·11-s − 1.93i·13-s − 1.59i·17-s − 0.857·19-s − 1.19i·23-s + (−0.122 + 0.992i)25-s − 0.371·29-s + 0.186·31-s + (0.283 − 0.250i)35-s + 1.75i·37-s + 1.08·41-s − 0.903i·43-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386393941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386393941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.48 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 + 6.96iT - 13T^{2} \) |
| 17 | \( 1 + 6.57iT - 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 10.7iT - 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 + 5.92iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 1.03iT - 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4.77iT - 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 + 4.26iT - 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 2.18T + 89T^{2} \) |
| 97 | \( 1 - 3.73iT - 97T^{2} \) |
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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
−8.092446527212452456112183543996, −7.28955932448936242347690690285, −6.52605190633038829966463412397, −5.76687976336322241186034275919, −4.91399870256452547995194947829, −4.32180242054984777899521558744, −3.42808077680690079459848667279, −2.62568968403820539755582098684, −1.23546059007034874155736715543, −0.41059636260877588406380433323,
1.38991060656235092260705503681, 2.10548156916025283416440357154, 3.55751048888563115503897577845, 4.04929919110808367131523028432, 4.36519840383402794786677195894, 5.96602426339193249601308705186, 6.43665308430958974485765246377, 6.99189219220116900669689199996, 7.66831872325400120732765199855, 8.553467641128741319277752981869
