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
−10.52352851755957055752416986782, −9.883198758484118356624002308057, −8.728325456624717991025874677742, −7.85647807859131432984202742773, −7.25579933366878470383595908214, −6.12341023524260049583606422742, −4.67158503965443930723474456491, −3.38132072358938026634034420409, −2.14491423181043650807699218974, −0.33146874134956456652030888190,
1.91571322044274871465573841039, 3.06487019702015541706366934730, 5.14546795670901539493250297099, 5.71933903954959306959651009087, 6.88051959170362657582616303692, 7.995554969902916268290509520719, 8.390017058915137707503987597230, 9.671788220804894441979783463434, 10.04205191128283796455569647679, 11.29886724686459558575423933126