| L(s) = 1 | + (1 − i)2-s − 2i·4-s + (−8 + 8i)7-s + (−2 − 2i)8-s + 4·11-s + (3 + 3i)13-s + 16i·14-s − 4·16-s + (−19 + 19i)17-s + 8i·19-s + (4 − 4i)22-s + (20 + 20i)23-s + 6·26-s + (16 + 16i)28-s + 38i·29-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.5i)2-s − 0.5i·4-s + (−1.14 + 1.14i)7-s + (−0.250 − 0.250i)8-s + 0.363·11-s + (0.230 + 0.230i)13-s + 1.14i·14-s − 0.250·16-s + (−1.11 + 1.11i)17-s + 0.421i·19-s + (0.181 − 0.181i)22-s + (0.869 + 0.869i)23-s + 0.230·26-s + (0.571 + 0.571i)28-s + 1.31i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.298415465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298415465\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (8 - 8i)T - 49iT^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 169iT^{2} \) |
| 17 | \( 1 + (19 - 19i)T - 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (-20 - 20i)T + 529iT^{2} \) |
| 29 | \( 1 - 38iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 70T + 1.68e3T^{2} \) |
| 43 | \( 1 + (36 + 36i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 2.20e3iT^{2} \) |
| 53 | \( 1 + (17 + 17i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 92iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72T + 3.72e3T^{2} \) |
| 67 | \( 1 + (44 - 44i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 88T + 5.04e3T^{2} \) |
| 73 | \( 1 + (55 + 55i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24 - 24i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 26iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-57 + 57i)T - 9.40e3iT^{2} \) |
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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
−11.16735063093424290702195738466, −10.29698093800720513732218259981, −9.120781526493720330849209139236, −8.858544468615040273682033763957, −7.12654776820321573802488581568, −6.17550557877759553897974421321, −5.45469971713568904307878173902, −4.01692006462026777210756038754, −3.06481587553087848869698363139, −1.78827824898256082582368843122,
0.44305221038273263766191932166, 2.75910930754673675747905998697, 3.88639314077383374462759074766, 4.78053673542332195963079660501, 6.20086433134799526670751983998, 6.85815010741146095588493947741, 7.61179383011849509141596673671, 8.951584402564481257549442833166, 9.665430619152321039611415685836, 10.79785339661914097149225309578
