| L(s) = 1 | + 0.792·2-s + (−1.26 − 1.18i)3-s − 1.37·4-s + (−1 − 0.939i)6-s + (1.73 + 2i)7-s − 2.67·8-s + (0.186 + 2.99i)9-s + 2.52i·11-s + (1.73 + 1.62i)12-s + 4.10·13-s + (1.37 + 1.58i)14-s + 0.627·16-s − 4.37i·17-s + (0.147 + 2.37i)18-s + 3.46i·19-s + ⋯ |
| L(s) = 1 | + 0.560·2-s + (−0.728 − 0.684i)3-s − 0.686·4-s + (−0.408 − 0.383i)6-s + (0.654 + 0.755i)7-s − 0.944·8-s + (0.0620 + 0.998i)9-s + 0.761i·11-s + (0.499 + 0.469i)12-s + 1.13·13-s + (0.366 + 0.423i)14-s + 0.156·16-s − 1.06i·17-s + (0.0347 + 0.559i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24558 + 0.267451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24558 + 0.267451i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.26 + 1.18i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 4.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.74iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.74iT - 43T^{2} \) |
| 47 | \( 1 - 1.62iT - 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4.74iT - 67T^{2} \) |
| 71 | \( 1 + 0.294iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−11.27893973386951434456351657696, −10.13215368918649517531731327317, −8.986080049623880865629247571898, −8.283444232890841702687134976349, −7.14168921351229082324364932095, −6.11162504961364887084990581502, −5.20542715906639265936282341519, −4.63080033300410024921108059716, −3.02407665648199169150376965414, −1.36742944608551150808558186017,
0.838721301636811534203425048740, 3.44505402013636177258084803891, 4.12031015900314002493891349209, 5.09096746642310717974550415489, 5.85855569060209032245785755024, 6.88619713892968861990071842212, 8.411150596072789314934466097985, 8.959197507933461851996908948836, 10.09674901734308681202563343948, 11.04878445724701657613037114015
