L(s) = 1 | + (1.41 + 1.41i)3-s + (−2.44 − 2.44i)7-s + 1.00i·9-s + 3.46·11-s + (−2.82 + 2.82i)13-s + (4.89 − 4.89i)17-s + 3.46i·19-s − 6.92i·21-s + (2.44 − 2.44i)23-s + (2.82 − 2.82i)27-s + 6.92·29-s + 8i·31-s + (4.89 + 4.89i)33-s + (5.65 + 5.65i)37-s − 8.00·39-s + ⋯ |
L(s) = 1 | + (0.816 + 0.816i)3-s + (−0.925 − 0.925i)7-s + 0.333i·9-s + 1.04·11-s + (−0.784 + 0.784i)13-s + (1.18 − 1.18i)17-s + 0.794i·19-s − 1.51i·21-s + (0.510 − 0.510i)23-s + (0.544 − 0.544i)27-s + 1.28·29-s + 1.43i·31-s + (0.852 + 0.852i)33-s + (0.929 + 0.929i)37-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.213725872\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.213725872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.41 - 1.41i)T + 3iT^{2} \) |
| 7 | \( 1 + (2.44 + 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + (2.82 - 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.89 + 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + (-5.65 - 5.65i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (1.41 + 1.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.44 + 2.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (4.89 + 4.89i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (4.24 + 4.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (4.89 - 4.89i)T - 97iT^{2} \) |
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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
−9.596456386168913004736387194702, −8.884069700290609954239470879071, −7.946915594718240182106105252901, −6.90750555927624983053099078596, −6.52441875694774413668609363396, −5.05056538796602194373575042197, −4.21971174941834103553147409915, −3.48298164336257268587769784962, −2.76610166145124910365512118920, −1.00874585733281223962936529567,
1.10604654108227544125240131950, 2.47465068334792953234808505006, 3.00037927684341909200602022101, 4.13917922281190248177002127462, 5.52307344157389981271076363064, 6.17280698336509045584302422835, 7.08648441912215638840755160266, 7.80894882278951461613444579401, 8.532358729611312237537432941105, 9.322861758313010104073975198543