L(s) = 1 | + 1.72i·5-s + (−2.63 + 0.222i)7-s − 6.17i·11-s + 2.82i·13-s − 1.72i·17-s − 5.90·19-s − 1.54i·23-s + 2.01·25-s + 8.28·29-s + 4.62·31-s + (−0.384 − 4.55i)35-s − 2.24·37-s + 3.92i·41-s + 10.1i·43-s + 4.88·47-s + ⋯ |
L(s) = 1 | + 0.772i·5-s + (−0.996 + 0.0839i)7-s − 1.86i·11-s + 0.784i·13-s − 0.419i·17-s − 1.35·19-s − 0.322i·23-s + 0.402·25-s + 1.53·29-s + 0.831·31-s + (−0.0649 − 0.770i)35-s − 0.368·37-s + 0.613i·41-s + 1.54i·43-s + 0.713·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.055039907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055039907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.222i)T \) |
good | 5 | \( 1 - 1.72iT - 5T^{2} \) |
| 11 | \( 1 + 6.17iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 1.72iT - 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 1.54iT - 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 - 3.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 14.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.64iT - 67T^{2} \) |
| 71 | \( 1 + 5.00iT - 71T^{2} \) |
| 73 | \( 1 - 2.19iT - 73T^{2} \) |
| 79 | \( 1 - 9.55iT - 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 9.16iT - 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−8.672182852198867881270954973931, −8.014309263106639267037091429382, −6.91330314048820272029789656888, −6.29604120645192832296754560253, −6.13878221775430480406672348379, −4.81916467937666900711190603540, −3.94982634658155775597114018725, −2.96546350736797477737306584240, −2.66875701203548890734027180920, −0.976243964121298971844244382193,
0.35496980857737903510868717774, 1.70403341528665529493179221865, 2.64572077959719076071134752861, 3.70364935842498823530593964052, 4.57224455418605531503752946230, 5.06076137656160164732815124015, 6.19583960768583839448899204932, 6.70025363658418664458663847876, 7.52684440837012629655364149668, 8.295209146192539802659283341577