| L(s) = 1 | + 2.14·3-s + 2.61i·5-s + 1.58·9-s + 3.95i·11-s + 1.08i·13-s + 5.59i·15-s − 0.317i·17-s + 5.16·19-s − 2.31i·23-s − 1.82·25-s − 3.02·27-s + 6.82·29-s − 6.05·31-s + 8.47i·33-s + 4·37-s + ⋯ |
| L(s) = 1 | + 1.23·3-s + 1.16i·5-s + 0.528·9-s + 1.19i·11-s + 0.300i·13-s + 1.44i·15-s − 0.0768i·17-s + 1.18·19-s − 0.483i·23-s − 0.365·25-s − 0.582·27-s + 1.26·29-s − 1.08·31-s + 1.47i·33-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.687763881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687763881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 3.95iT - 11T^{2} \) |
| 13 | \( 1 - 1.08iT - 13T^{2} \) |
| 17 | \( 1 + 0.317iT - 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 2.31iT - 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.23iT - 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.41iT - 61T^{2} \) |
| 67 | \( 1 + 3.27iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 14.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.91iT - 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 5.99iT - 89T^{2} \) |
| 97 | \( 1 + 9.23iT - 97T^{2} \) |
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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
−8.953555391155867233238912478721, −8.016580864782143821157969064161, −7.45351698798317327602004751892, −6.89101257870796313253024011841, −6.05956072308572051635491868854, −4.86663860116045450233149676610, −4.02785709736989374889857766809, −3.01425763608413461662968448422, −2.65971762806391538109163221990, −1.58555692437347888327444806492,
0.69655544460480092722047955769, 1.75844855584303169486035714766, 3.01154064161068046486246859782, 3.48089091886406156716039238064, 4.54447472091886660190839299335, 5.39569692284735241912765964399, 6.05637138743378388860998793726, 7.32410199646190380255004872622, 7.964937459304379987848384500635, 8.526948402026597512147594441280
