| L(s) = 1 | + 2.82i·5-s + 2.82·7-s − 4i·11-s − 5.65i·13-s + 2·17-s − 4i·19-s + 5.65·23-s − 3.00·25-s + 2.82i·29-s + 8.48·31-s + 8.00i·35-s − 10·41-s + 12i·43-s + 5.65·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s + 1.06·7-s − 1.20i·11-s − 1.56i·13-s + 0.485·17-s − 0.917i·19-s + 1.17·23-s − 0.600·25-s + 0.525i·29-s + 1.52·31-s + 1.35i·35-s − 1.56·41-s + 1.82i·43-s + 0.825·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899473512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899473512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.08082160548055394669298808759, −8.783761315203653913527569818280, −8.127037837729327014122770089733, −7.37903711120285509448947027289, −6.47923965040494066937806798261, −5.56922694993706530252994530735, −4.73583066904685168062301457370, −3.18874602338073949212864779184, −2.84198226003781938157387398238, −1.02599418537207740028973219272,
1.28320948559777165490831403198, 2.08086361289812142499447582937, 3.93630362445975200210690591408, 4.75629565840667196882553723715, 5.16977221101405703143727365532, 6.51534251112862772691056997974, 7.42958213784013402493336879992, 8.277392892570955120413648515043, 8.921324263623792735755783259392, 9.671223497338860981189015563234
