| L(s) = 1 | + i·2-s − 4-s − 1.01·5-s + (0.141 − 2.64i)7-s − i·8-s − 1.01i·10-s − 5.07i·11-s + 5.45i·13-s + (2.64 + 0.141i)14-s + 16-s − 6.19·17-s + 7.96i·19-s + 1.01·20-s + 5.07·22-s + 0.835i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.456·5-s + (0.0533 − 0.998i)7-s − 0.353i·8-s − 0.322i·10-s − 1.53i·11-s + 1.51i·13-s + (0.706 + 0.0377i)14-s + 0.250·16-s − 1.50·17-s + 1.82i·19-s + 0.228·20-s + 1.08·22-s + 0.174i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4169806233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4169806233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.141 + 2.64i)T \) |
| good | 5 | \( 1 + 1.01T + 5T^{2} \) |
| 11 | \( 1 + 5.07iT - 11T^{2} \) |
| 13 | \( 1 - 5.45iT - 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 - 7.96iT - 19T^{2} \) |
| 23 | \( 1 - 0.835iT - 23T^{2} \) |
| 29 | \( 1 - 8.91iT - 29T^{2} \) |
| 31 | \( 1 - 3.97iT - 31T^{2} \) |
| 37 | \( 1 + 2.64T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 - 5.67T + 43T^{2} \) |
| 47 | \( 1 - 0.563T + 47T^{2} \) |
| 53 | \( 1 + 2.19iT - 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 + 7.94iT - 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 6.20iT - 71T^{2} \) |
| 73 | \( 1 - 7.05iT - 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 + 2.78iT - 97T^{2} \) |
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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
−10.22836175878567294033152861364, −9.114337902786441824751116229354, −8.523122809112734309243876351818, −7.72928386574928860188700882772, −6.81927776580515958067358226716, −6.27314198179820861703573584829, −5.11365094992438173623822136010, −4.04833389490433320476037165456, −3.53039117851586972743464016356, −1.55445873505594662591945532676,
0.17954914318116927941039594637, 2.15461116268021019294450992368, 2.75939629002308664853311470378, 4.20783121390404435069828380658, 4.85514206187504859933073319912, 5.87472995131015628243212248708, 7.02185877273648368008756911960, 7.907376566323412669002759517599, 8.739427521607537186250459233602, 9.489666470191019876539486761067
