| L(s) = 1 | + (−0.706 − 0.360i)2-s + (0.668 + 1.31i)3-s + (−0.805 − 1.10i)4-s + (−0.0570 + 2.23i)5-s − 1.16i·6-s + (−0.442 + 2.79i)7-s + (0.418 + 2.64i)8-s + (0.490 − 0.675i)9-s + (0.845 − 1.55i)10-s + (1.59 + 2.19i)11-s + (0.915 − 1.79i)12-s + (−0.417 − 0.819i)13-s + (1.31 − 1.81i)14-s + (−2.96 + 1.41i)15-s + (−0.191 + 0.589i)16-s + (0.225 + 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.499 − 0.254i)2-s + (0.385 + 0.756i)3-s + (−0.402 − 0.554i)4-s + (−0.0255 + 0.999i)5-s − 0.476i·6-s + (−0.167 + 1.05i)7-s + (0.147 + 0.933i)8-s + (0.163 − 0.225i)9-s + (0.267 − 0.493i)10-s + (0.480 + 0.661i)11-s + (0.264 − 0.518i)12-s + (−0.115 − 0.227i)13-s + (0.352 − 0.485i)14-s + (−0.766 + 0.366i)15-s + (−0.0478 + 0.147i)16-s + (0.0545 + 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.746861 + 0.467410i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746861 + 0.467410i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.0570 - 2.23i)T \) |
| 31 | \( 1 + (3.59 + 4.24i)T \) |
| good | 2 | \( 1 + (0.706 + 0.360i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.668 - 1.31i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.442 - 2.79i)T + (-6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.59 - 2.19i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.417 + 0.819i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.225 - 1.42i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (0.424 - 0.137i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.31 + 0.841i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.54 + 7.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.175 + 0.175i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.04 - 3.22i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.91 + 1.48i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (3.40 + 6.67i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.74 + 0.592i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-13.4 - 4.37i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + (1.20 - 1.20i)T - 67iT^{2} \) |
| 71 | \( 1 + (-5.46 - 3.97i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.86 - 11.7i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.21 - 3.06i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-13.4 - 6.83i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.24 - 4.53i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.315 - 1.99i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.18092445818283396822921011056, −11.85767998614920935609099173123, −10.86759902367080846555575873178, −9.819885333175724611330536925665, −9.428774628525262990564960231869, −8.306347933212097778990446964888, −6.71467781347072564997928727794, −5.45113944976922311702554989602, −3.95539837953763517701545671366, −2.37206704813008061322044001065,
1.10565679726837749507030148811, 3.58496130908101778863964404886, 4.88985852652607809344438918376, 6.88402219679991511452785659196, 7.56449606551884072190512156298, 8.611030923444487797056163588403, 9.284176402230895052377374427498, 10.63967989497295540613186632127, 12.07154174860230219347532583006, 13.06618153812660555133080510815
