| L(s) = 1 | − 2.82i·5-s + i·11-s + 0.635i·13-s − 2.89·17-s + 4.89i·19-s + 0.635·23-s − 3.00·25-s − 6.29i·29-s − 2.19·31-s + 6.92i·37-s + 1.10·41-s + 0.898i·43-s + 6.29·47-s − 7·49-s − 4.09i·53-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s + 0.301i·11-s + 0.176i·13-s − 0.703·17-s + 1.12i·19-s + 0.132·23-s − 0.600·25-s − 1.16i·29-s − 0.393·31-s + 1.13i·37-s + 0.171·41-s + 0.137i·43-s + 0.917·47-s − 49-s − 0.563i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031653268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031653268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 0.635iT - 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 0.635T + 23T^{2} \) |
| 29 | \( 1 + 6.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898iT - 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 4.09iT - 53T^{2} \) |
| 59 | \( 1 - 13.7iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 9.79iT - 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.251408074926295213917067071000, −7.61182890133763288009616056980, −6.76540121118684956505727264191, −5.93062076370992577380474448861, −5.34882091978154721824833902670, −4.41682613300528193494594913883, −4.13339883375354333188547415121, −2.90357023753256584848419572247, −1.86410942490445144415504179899, −1.05500268496502909291381261530,
0.27412587009344105598489200721, 1.77932992077777203492731166050, 2.74434471142083341202838692865, 3.24852564257003489086148419126, 4.18250692991937241148947002264, 5.05207637231783145087812128872, 5.86261368238343526355754529642, 6.64592350980053799089610958689, 7.07627380368831004704131945435, 7.70074937709520328702598810132
