L(s) = 1 | + 2.18i·3-s + 0.351i·5-s − 3.15i·7-s − 1.75·9-s + 3.05i·11-s − 2.45·13-s − 0.767·15-s + (0.112 − 4.12i)17-s + 2.18·19-s + 6.88·21-s + 0.185i·23-s + 4.87·25-s + 2.71i·27-s + 4.16i·29-s − 6.26i·31-s + ⋯ |
L(s) = 1 | + 1.25i·3-s + 0.157i·5-s − 1.19i·7-s − 0.585·9-s + 0.922i·11-s − 0.680·13-s − 0.198·15-s + (0.0272 − 0.999i)17-s + 0.500·19-s + 1.50·21-s + 0.0387i·23-s + 0.975·25-s + 0.521i·27-s + 0.774i·29-s − 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.772732322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.772732322\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-0.112 + 4.12i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 2.18iT - 3T^{2} \) |
| 5 | \( 1 - 0.351iT - 5T^{2} \) |
| 7 | \( 1 + 3.15iT - 7T^{2} \) |
| 11 | \( 1 - 3.05iT - 11T^{2} \) |
| 13 | \( 1 + 2.45T + 13T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 0.185iT - 23T^{2} \) |
| 29 | \( 1 - 4.16iT - 29T^{2} \) |
| 31 | \( 1 + 6.26iT - 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 6.35iT - 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 - 0.937T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 61 | \( 1 + 1.52iT - 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 16.4iT - 97T^{2} \) |
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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
−8.875910203846384204441595037871, −7.73817220009114881324962223977, −7.18708503317717269856930834630, −6.62629546987070922862441382901, −5.18007834395745969626396610838, −4.89415698410428894088348435831, −4.10248506734983037235732358091, −3.41726273067407058145996075672, −2.43068877468862823155282746268, −0.951810795980915826763539971551,
0.63304873000287577761371212379, 1.75666354860780592423843462448, 2.51375084553376581937917489089, 3.37508201161867812336793929561, 4.61198960708531111994936612690, 5.61923681475499698924626818391, 5.99050099645119848402459181403, 6.84707135114915656533225874693, 7.50566542101952084510318171620, 8.343377990665935254105877648891