| L(s) = 1 | + 0.806i·3-s − 0.0998i·7-s + 2.34·9-s + 5.37·11-s − 1.73i·13-s + 4.30i·17-s + 3.44·19-s + 0.0805·21-s − 1.32i·23-s + 4.31i·27-s + 1.62·29-s − 1.63·31-s + 4.33i·33-s − 6.20i·37-s + 1.40·39-s + ⋯ |
| L(s) = 1 | + 0.465i·3-s − 0.0377i·7-s + 0.783·9-s + 1.61·11-s − 0.482i·13-s + 1.04i·17-s + 0.791·19-s + 0.0175·21-s − 0.275i·23-s + 0.830i·27-s + 0.301·29-s − 0.293·31-s + 0.754i·33-s − 1.01i·37-s + 0.224·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.454534687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.454534687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.806iT - 3T^{2} \) |
| 7 | \( 1 + 0.0998iT - 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 4.30iT - 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 + 1.32iT - 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 6.20iT - 37T^{2} \) |
| 43 | \( 1 + 7.87iT - 43T^{2} \) |
| 47 | \( 1 - 9.86iT - 47T^{2} \) |
| 53 | \( 1 - 0.388iT - 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 + 9.53iT - 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 3.26iT - 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 + 6.27iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.08iT - 97T^{2} \) |
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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
−8.655015281055657055162629398875, −7.66364559208058462698319708916, −7.05396284555484844662530225402, −6.25221978138816839908349646480, −5.55947647412850174878119995191, −4.51440353520557782593236420576, −3.95475550947500341069636163348, −3.27280004039343095067056165034, −1.89842402784439809095544006290, −0.993426680301567054795651083511,
0.962528193543742864490219158425, 1.67086017440985590156074427736, 2.84272621949021104709557067831, 3.86320563938591552648382393032, 4.49685995008680350231924472910, 5.43116294023385627535330392366, 6.40281158328682755659311737667, 6.95004261976266701818663076090, 7.41705446136897162986138313792, 8.383350664492254963065531926054
