| L(s) = 1 | + 0.0864i·7-s + 0.913·11-s − 2.62i·13-s − 2.08i·17-s − 4.93·19-s + 8.47i·23-s + 2.39·29-s + 3.62·31-s − 5.85i·37-s − 6.64·41-s + 8.24i·43-s − 2.68i·47-s + 6.99·49-s − 5.73i·53-s + 12.3·59-s + ⋯ |
| L(s) = 1 | + 0.0326i·7-s + 0.275·11-s − 0.728i·13-s − 0.506i·17-s − 1.13·19-s + 1.76i·23-s + 0.445·29-s + 0.651·31-s − 0.962i·37-s − 1.03·41-s + 1.25i·43-s − 0.392i·47-s + 0.998·49-s − 0.787i·53-s + 1.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.821971593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821971593\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 0.0864iT - 7T^{2} \) |
| 11 | \( 1 - 0.913T + 11T^{2} \) |
| 13 | \( 1 + 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 2.08iT - 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 - 8.47iT - 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + 5.85iT - 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 8.24iT - 43T^{2} \) |
| 47 | \( 1 + 2.68iT - 47T^{2} \) |
| 53 | \( 1 + 5.73iT - 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 6.16iT - 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.31iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 6.07iT - 83T^{2} \) |
| 89 | \( 1 + 8.13T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−7.86016523101154801789090335180, −7.03832625238680315570861566395, −6.46172357572007415983995697414, −5.61530449389149773368470751217, −5.08925496586377511130375452103, −4.16113057035459589422276546434, −3.47310418290975718616008135052, −2.62437178286639209039982698159, −1.70227411346612584715228358013, −0.58696420312106128655798756174,
0.75776949097187387591155256240, 1.93601251434315981526591298913, 2.60155688874957401103814421705, 3.69936042011946549955127218488, 4.34687489456415856487562112287, 4.93230833119049945947734471836, 5.95475732217475367625455256770, 6.62596491635412503117545990968, 6.90975739076113583933875162068, 8.081128730707212177043930088592
