| L(s) = 1 | + 1.30i·2-s + i·3-s + 0.302·4-s − 1.30·6-s − i·7-s + 3i·8-s − 9-s − 3·11-s + 0.302i·12-s + 4.60i·13-s + 1.30·14-s − 3.30·16-s + 2.60i·17-s − 1.30i·18-s + 0.605·19-s + ⋯ |
| L(s) = 1 | + 0.921i·2-s + 0.577i·3-s + 0.151·4-s − 0.531·6-s − 0.377i·7-s + 1.06i·8-s − 0.333·9-s − 0.904·11-s + 0.0874i·12-s + 1.27i·13-s + 0.348·14-s − 0.825·16-s + 0.631i·17-s − 0.307i·18-s + 0.138·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.324578 + 1.37493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324578 + 1.37493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 - 1.30iT - 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 2.60iT - 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 - 8.21iT - 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.39iT - 43T^{2} \) |
| 47 | \( 1 + 3.39iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 8.39iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 6.60iT - 73T^{2} \) |
| 79 | \( 1 + 6.81T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 15.2iT - 97T^{2} \) |
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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
−11.21719683906023119161274407852, −10.31874704359124635023695431045, −9.396796697623016261712292950456, −8.383045076922166270285061684555, −7.57894015741015216091566339209, −6.71601904965031689346119385666, −5.72733955661966399703542866562, −4.86006180053234722446616399071, −3.65662647170474264635971658041, −2.12758258962844642651920096441,
0.817349675956194043528202335645, 2.49657904822927266617452528311, 3.01035985076929984123651179510, 4.69078669539520149296832886637, 5.87959130281943471146517372703, 6.84636163051939418062356948804, 7.85707492403419472580481538524, 8.665058098138575599927104892283, 10.05091741283857856232682149880, 10.43652810613648470063161706234
