| L(s) = 1 | + i·3-s − 0.585i·5-s − 1.41·7-s − 9-s − 0.828i·11-s + 4.82i·13-s + 0.585·15-s − 0.828·17-s − 2.82i·19-s − 1.41i·21-s − 6.82·23-s + 4.65·25-s − i·27-s + 4.58i·29-s − 7.07·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.261i·5-s − 0.534·7-s − 0.333·9-s − 0.249i·11-s + 1.33i·13-s + 0.151·15-s − 0.200·17-s − 0.648i·19-s − 0.308i·21-s − 1.42·23-s + 0.931·25-s − 0.192i·27-s + 0.851i·29-s − 1.27·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3569014584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3569014584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 4.58iT - 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 1.17iT - 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 10.2iT - 53T^{2} \) |
| 59 | \( 1 - 9.65iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.17iT - 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−9.850419075281036045823010189883, −8.904865624385388404467974117115, −8.689232649563766310007177116609, −7.32996830203883565412282282439, −6.63082812985153838365314373840, −5.72889804071367522450887319899, −4.76819249081194432091444584039, −3.99760333820536496709684865622, −3.04379289422542601740046747468, −1.75381991205540140365448913973,
0.13184807619422130423007832543, 1.73903000302059566187873504149, 2.89973967448880990277102745805, 3.72251586255089631938478399289, 5.01412578952117796449248243849, 5.96991430587407052055022812463, 6.53261808786800638563227371289, 7.59164480557348329724746401077, 8.036754152603138823633037216992, 9.021520334584658519029126752337
