| L(s) = 1 | + 5-s − 0.454·7-s − 2.12·11-s − 4.88·13-s + 7.66·17-s − 0.330·19-s − 9.13·23-s + 25-s + 8.57·29-s − 1.54·31-s − 0.454·35-s − 37-s + 12.2·41-s − 4.57·43-s − 3.79·47-s − 6.79·49-s − 2.90·53-s − 2.12·55-s + 8.12·59-s + 10.7·61-s − 4.88·65-s + 3.84·67-s + 13.2·71-s + 4.12·73-s + 0.966·77-s − 9.94·79-s − 8.55·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.171·7-s − 0.640·11-s − 1.35·13-s + 1.86·17-s − 0.0759·19-s − 1.90·23-s + 0.200·25-s + 1.59·29-s − 0.277·31-s − 0.0768·35-s − 0.164·37-s + 1.90·41-s − 0.698·43-s − 0.553·47-s − 0.970·49-s − 0.399·53-s − 0.286·55-s + 1.05·59-s + 1.37·61-s − 0.605·65-s + 0.470·67-s + 1.57·71-s + 0.482·73-s + 0.110·77-s − 1.11·79-s − 0.938·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779354250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779354250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 0.454T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 + 0.330T + 19T^{2} \) |
| 23 | \( 1 + 9.13T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 3.84T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 4.12T + 73T^{2} \) |
| 79 | \( 1 + 9.94T + 79T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−7.959844375327725641876934250642, −7.40087330966937627518343558872, −6.54913528606288728279463184738, −5.79487269740754251313017962802, −5.23544607595406626197601844370, −4.49420510856595049682830332493, −3.49557356938706911158822885513, −2.68464232716354433496291146215, −1.95021264637253502943304211355, −0.67025161489581197336871275564,
0.67025161489581197336871275564, 1.95021264637253502943304211355, 2.68464232716354433496291146215, 3.49557356938706911158822885513, 4.49420510856595049682830332493, 5.23544607595406626197601844370, 5.79487269740754251313017962802, 6.54913528606288728279463184738, 7.40087330966937627518343558872, 7.959844375327725641876934250642
