| L(s) = 1 | − 1.02i·2-s + (−6.61 − 3.82i)3-s + 14.9·4-s + (−23.6 − 13.6i)5-s + (−3.93 + 6.81i)6-s + (−37.2 + 21.4i)7-s − 31.8i·8-s + (−11.2 − 19.5i)9-s + (−14.0 + 24.3i)10-s − 144.·11-s + (−98.8 − 57.0i)12-s + (−62.0 − 107. i)13-s + (22.1 + 38.3i)14-s + (104. + 180. i)15-s + 206.·16-s + (−32.5 − 56.4i)17-s + ⋯ |
| L(s) = 1 | − 0.257i·2-s + (−0.735 − 0.424i)3-s + 0.933·4-s + (−0.944 − 0.545i)5-s + (−0.109 + 0.189i)6-s + (−0.759 + 0.438i)7-s − 0.497i·8-s + (−0.139 − 0.241i)9-s + (−0.140 + 0.243i)10-s − 1.19·11-s + (−0.686 − 0.396i)12-s + (−0.367 − 0.636i)13-s + (0.112 + 0.195i)14-s + (0.463 + 0.802i)15-s + 0.805·16-s + (−0.112 − 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0997805 - 0.628999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0997805 - 0.628999i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (721. + 1.70e3i)T \) |
| good | 2 | \( 1 + 1.02iT - 16T^{2} \) |
| 3 | \( 1 + (6.61 + 3.82i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (23.6 + 13.6i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (37.2 - 21.4i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 144.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (62.0 + 107. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (32.5 + 56.4i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-410. - 237. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-411. + 712. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-128. + 73.9i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-194. + 336. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-191. - 110. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 1.10e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 3.55e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-894. + 1.54e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 - 4.90e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + (846. - 488. i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.68e3 + 4.64e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-4.19 + 2.42i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (7.57e3 - 4.37e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-5.74e3 - 9.94e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-3.21e3 + 5.57e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (7.58e3 + 4.37e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 1.20e4T + 8.85e7T^{2} \) |
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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
−15.08605556476609623975694115034, −12.86121539218575267579665487665, −12.26155448617979505608255261404, −11.38720487503992398260412552102, −10.06559697106946139767199320162, −8.133814427549034959196793122980, −6.83715225981518556844452743279, −5.43829032568276767714109873405, −3.03584792259186730200285557874, −0.42042684381699165918059579790,
3.11619861714171264565290077871, 5.21789415533589924985944723310, 6.83238643104333383507601484318, 7.75819721408389446619100363226, 10.02668206845189231066467903756, 11.12138739747332325971201322043, 11.71930826033962754585090247401, 13.38127881943105762004043845465, 15.03529109167562878123018882442, 15.93656036289247258411370837106
