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
\(L(1)\) |
\(\approx\) |
\(2.55396 + 0.634065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55396 + 0.634065i\) |
\(L(\frac{3}{2})\) |
|
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
−11.45975999412532466250009921785, −10.18413320571294103166756396745, −9.216896076703753399728806394361, −7.976017164902637711128153014159, −7.15448594092494407024804453585, −6.40412779019830542028468426441, −5.34280541093949137170313142818, −4.05935861119181829018965639161, −3.57297918303824143140678425069, −1.69492095634703892192107754173,
1.63074218389756107788253821593, 2.91858183834966524370532660914, 4.08369133780227625738815326125, 5.07276299901593456960217751829, 6.23659790698690867258527222094, 6.70383656298332919249825860080, 8.228763989252031851458241720097, 9.215245200596991221119095223102, 10.05449145463499150584180947663, 11.14613601025898991266077944812