| L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 2i·13-s − 14-s + 16-s + 6i·17-s + 19-s − 6i·23-s − 2·26-s − i·28-s − 6·29-s + 5·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s + 1.45i·17-s + 0.229·19-s − 1.25i·23-s − 0.392·26-s − 0.188i·28-s − 1.11·29-s + 0.898·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102353884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102353884\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−9.836622120398146107072901775391, −8.944114795050053596617946401790, −8.325396380328993940725518319958, −7.57982628729479529230328223731, −6.44522956847503204492409833762, −6.10652183940650925047503430102, −4.91882461052074495254691884640, −4.17832460918002345553434728074, −2.98293020204661445779113799761, −1.55390017061958050955214855550,
0.45994623988403635555091252231, 1.88115637931122366117318753030, 3.09075971304660146306979084986, 3.86161969296874436345554460870, 5.03583953417591464402076865678, 5.64457833596253544159827872758, 7.03961768579907224725114094771, 7.58191430796751067552709408838, 8.659931308072713457773860968380, 9.389517595187708570774154348231
