| L(s) = 1 | + 16.9·11-s + 2·17-s − 16.9·19-s + 25·25-s + 46·41-s − 84.8·43-s + 49·49-s + 84.8·59-s + 118.·67-s + 142·73-s − 50.9·83-s + 146·89-s + 94·97-s − 118.·107-s − 98·113-s + ⋯ |
| L(s) = 1 | + 1.54·11-s + 0.117·17-s − 0.893·19-s + 25-s + 1.12·41-s − 1.97·43-s + 0.999·49-s + 1.43·59-s + 1.77·67-s + 1.94·73-s − 0.613·83-s + 1.64·89-s + 0.969·97-s − 1.11·107-s − 0.867·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.115730911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115730911\) |
| \(L(2)\) |
|
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 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 2T + 289T^{2} \) |
| 19 | \( 1 + 16.9T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 46T + 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 142T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 146T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−9.510380022083280560984996594734, −8.834946462457528634002407209240, −8.091018542341745682765678841086, −6.87795765864034954514253310386, −6.47079769670291524895213578533, −5.33361850851378069081623965878, −4.29622359481420015054870779451, −3.49658909901853304742181034459, −2.15570829318574668579728710231, −0.910793758751232266365453662713,
0.910793758751232266365453662713, 2.15570829318574668579728710231, 3.49658909901853304742181034459, 4.29622359481420015054870779451, 5.33361850851378069081623965878, 6.47079769670291524895213578533, 6.87795765864034954514253310386, 8.091018542341745682765678841086, 8.834946462457528634002407209240, 9.510380022083280560984996594734
