| L(s) = 1 | + (−1.32 + 0.5i)2-s + (1.50 − 1.32i)4-s + 2.64i·7-s + (−1.32 + 2.50i)8-s − 5.29·11-s + (−1.32 − 3.50i)14-s + (0.500 − 3.96i)16-s + (7.00 − 2.64i)22-s + 8i·23-s − 5·25-s + (3.50 + 3.96i)28-s − 2i·29-s + (1.32 + 5.50i)32-s + 10.5i·37-s − 12·43-s + (−7.93 + 7.00i)44-s + ⋯ |
| L(s) = 1 | + (−0.935 + 0.353i)2-s + (0.750 − 0.661i)4-s + 0.999i·7-s + (−0.467 + 0.883i)8-s − 1.59·11-s + (−0.353 − 0.935i)14-s + (0.125 − 0.992i)16-s + (1.49 − 0.564i)22-s + 1.66i·23-s − 25-s + (0.661 + 0.749i)28-s − 0.371i·29-s + (0.233 + 0.972i)32-s + 1.73i·37-s − 1.82·43-s + (−1.19 + 1.05i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106712 + 0.429829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106712 + 0.429829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 16iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.29886724686459558575423933126, −10.04205191128283796455569647679, −9.671788220804894441979783463434, −8.390017058915137707503987597230, −7.995554969902916268290509520719, −6.88051959170362657582616303692, −5.71933903954959306959651009087, −5.14546795670901539493250297099, −3.06487019702015541706366934730, −1.91571322044274871465573841039,
0.33146874134956456652030888190, 2.14491423181043650807699218974, 3.38132072358938026634034420409, 4.67158503965443930723474456491, 6.12341023524260049583606422742, 7.25579933366878470383595908214, 7.85647807859131432984202742773, 8.728325456624717991025874677742, 9.883198758484118356624002308057, 10.52352851755957055752416986782
