| L(s) = 1 | + (2.82 − i)3-s + (7.00 − 5.65i)9-s + 14i·11-s + 33.9i·17-s + 16.9·19-s + 25·25-s + (14.1 − 23.0i)27-s + (14 + 39.5i)33-s − 67.8i·41-s + 84.8·43-s − 49·49-s + (33.9 + 96i)51-s + (48 − 16.9i)57-s + 82i·59-s + 118.·67-s + ⋯ |
| L(s) = 1 | + (0.942 − 0.333i)3-s + (0.777 − 0.628i)9-s + 1.27i·11-s + 1.99i·17-s + 0.893·19-s + 25-s + (0.523 − 0.851i)27-s + (0.424 + 1.19i)33-s − 1.65i·41-s + 1.97·43-s − 0.999·49-s + (0.665 + 1.88i)51-s + (0.842 − 0.297i)57-s + 1.38i·59-s + 1.77·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.713505757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.713505757\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 + i)T \) |
| good | 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 14iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 33.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.9T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 84.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 82iT - 3.48e3T^{2} \) |
| 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 | \( 1 + 158iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 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
−10.07762701164365665912290449968, −9.225316569848696207496495115851, −8.478719742139602842851378062778, −7.57143696047222344454862934824, −6.94029189952588883552514372375, −5.83815802319003284027986191977, −4.49799617493307901531764614797, −3.64266858952533784618349501352, −2.39917421385043750282690161181, −1.36875101064176717597572361048,
0.950199002863151206450412548067, 2.70411194709296404025987565066, 3.28786986920726198447903194643, 4.58491949272348302023670203975, 5.45073797576647302856782005838, 6.76040856783115788878324961051, 7.63495168565673535699088155094, 8.427774916932308130465791825870, 9.292164242521792348664134428133, 9.780569734774225477245112981792
