| L(s) = 1 | + 0.302i·3-s + 0.697i·7-s + 2.90·9-s − 11-s + 1.60i·13-s − 4.30i·17-s − 19-s − 0.211·21-s − 3.90i·23-s + 1.78i·27-s + 1.69·29-s − 1.60·31-s − 0.302i·33-s − 7.60i·37-s − 0.486·39-s + ⋯ |
| L(s) = 1 | + 0.174i·3-s + 0.263i·7-s + 0.969·9-s − 0.301·11-s + 0.445i·13-s − 1.04i·17-s − 0.229·19-s − 0.0460·21-s − 0.814i·23-s + 0.344i·27-s + 0.315·29-s − 0.288·31-s − 0.0527i·33-s − 1.25i·37-s − 0.0778·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927408012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927408012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.302iT - 3T^{2} \) |
| 7 | \( 1 - 0.697iT - 7T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 + 4.30iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 3.90iT - 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 7.60iT - 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 - 5.60iT - 47T^{2} \) |
| 53 | \( 1 + 6.51iT - 53T^{2} \) |
| 59 | \( 1 + 5.60T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 1.90iT - 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 6.90iT - 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 7.11iT - 97T^{2} \) |
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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
−8.344573920033322795387743962885, −7.41920510504514355211102905922, −7.01632203360154801084010968984, −6.11450049089458274239051017693, −5.28328408632907170847424162540, −4.51471558782671196501774358900, −3.89231819570873533968659474575, −2.75194241620424771282725680812, −1.96442615971601037506361401382, −0.64678935939778564069675681632,
0.991850641403497964335931857388, 1.89581907513576909277294624701, 3.01061836199903331082567954369, 3.93310200131760935683869340163, 4.58187941901437908251569839103, 5.52066244859662052465485121896, 6.27857814055068653664936550357, 6.99273201581962014220671146853, 7.77646047716154129824682257465, 8.176175677992336728879487421953
