L(s) = 1 | + i·2-s + i·3-s − 4-s + 3.51i·5-s − 6-s − i·8-s − 9-s − 3.51·10-s + (−2.26 + 2.42i)11-s − i·12-s − 5.40·13-s − 3.51·15-s + 16-s − 2.00·17-s − i·18-s − 4.64·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.57i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.11·10-s + (−0.683 + 0.730i)11-s − 0.288i·12-s − 1.50·13-s − 0.907·15-s + 0.250·16-s − 0.485·17-s − 0.235i·18-s − 1.06·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4149591223\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4149591223\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.26 - 2.42i)T \) |
good | 5 | \( 1 - 3.51iT - 5T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 - 8.62iT - 29T^{2} \) |
| 31 | \( 1 - 5.01iT - 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 0.994T + 41T^{2} \) |
| 43 | \( 1 + 9.82iT - 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 0.0847iT - 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 7.10T + 73T^{2} \) |
| 79 | \( 1 - 8.52iT - 79T^{2} \) |
| 83 | \( 1 + 3.88T + 83T^{2} \) |
| 89 | \( 1 - 3.47iT - 89T^{2} \) |
| 97 | \( 1 - 8.90iT - 97T^{2} \) |
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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
−9.275047766448444853000116021954, −8.532456248552779955135937117792, −7.52950822888332950414671896884, −6.91166294658034876062596615961, −6.69081933539253264151043224557, −5.32830481073187308448667190423, −4.93455378015691922685095353702, −3.89512105608812690396973332885, −2.92249357767559940546631746784, −2.25396647298749869914124850719,
0.14489624252619195506748352085, 0.980800810533718307359948797383, 2.19633026747684241622918382739, 2.86229544066144882240005778682, 4.34857203872146650586167699811, 4.72957228182287485335031333088, 5.57653527908555732043570131001, 6.37705550719897375334762062625, 7.64852346938663848507721121416, 8.041346055206757617484660350310