| L(s) = 1 | − 0.786·3-s + 3.29·5-s + 2.08·7-s − 2.38·9-s − 1.29·11-s + 1.21·13-s − 2.59·15-s + 4.08·17-s + 19-s − 1.63·21-s + 8.95·23-s + 5.87·25-s + 4.23·27-s − 9.38·29-s + 1.02·33-s + 6.87·35-s − 2·37-s − 0.954·39-s + 3.57·41-s − 7.72·43-s − 7.85·45-s − 9.46·47-s − 2.65·49-s − 3.21·51-s − 11.9·53-s − 4.27·55-s − 0.786·57-s + ⋯ |
| L(s) = 1 | − 0.454·3-s + 1.47·5-s + 0.787·7-s − 0.793·9-s − 0.391·11-s + 0.336·13-s − 0.669·15-s + 0.990·17-s + 0.229·19-s − 0.357·21-s + 1.86·23-s + 1.17·25-s + 0.814·27-s − 1.74·29-s + 0.177·33-s + 1.16·35-s − 0.328·37-s − 0.152·39-s + 0.558·41-s − 1.17·43-s − 1.17·45-s − 1.38·47-s − 0.379·49-s − 0.449·51-s − 1.64·53-s − 0.576·55-s − 0.104·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458668333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458668333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.786T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−11.46041065289235852729454214063, −10.91552549995755363480443255575, −9.869133929594604236173049500670, −9.015355166181452972281499650357, −7.946775557002075432635587888010, −6.63093008896007893457608729078, −5.52399931717913474337243460244, −5.13009513375226831578289402134, −3.05986132748281541374303515005, −1.56727195113053216494868612301,
1.56727195113053216494868612301, 3.05986132748281541374303515005, 5.13009513375226831578289402134, 5.52399931717913474337243460244, 6.63093008896007893457608729078, 7.946775557002075432635587888010, 9.015355166181452972281499650357, 9.869133929594604236173049500670, 10.91552549995755363480443255575, 11.46041065289235852729454214063
