| L(s) = 1 | − 0.0333·2-s − 1.99·4-s − 4.39·5-s − 2.75·7-s + 0.133·8-s + 0.146·10-s − 4.02·11-s − 2.03·13-s + 0.0920·14-s + 3.99·16-s − 3.07·17-s − 2.83·19-s + 8.79·20-s + 0.134·22-s − 8.64·23-s + 14.3·25-s + 0.0680·26-s + 5.51·28-s − 1.87·29-s − 2.12·31-s − 0.400·32-s + 0.102·34-s + 12.1·35-s + 0.623·37-s + 0.0944·38-s − 0.587·40-s − 3.32·41-s + ⋯ |
| L(s) = 1 | − 0.0235·2-s − 0.999·4-s − 1.96·5-s − 1.04·7-s + 0.0471·8-s + 0.0464·10-s − 1.21·11-s − 0.565·13-s + 0.0246·14-s + 0.998·16-s − 0.746·17-s − 0.649·19-s + 1.96·20-s + 0.0286·22-s − 1.80·23-s + 2.87·25-s + 0.0133·26-s + 1.04·28-s − 0.347·29-s − 0.381·31-s − 0.0707·32-s + 0.0176·34-s + 2.05·35-s + 0.102·37-s + 0.0153·38-s − 0.0928·40-s − 0.519·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07964054359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07964054359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0333T + 2T^{2} \) |
| 5 | \( 1 + 4.39T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 + 8.64T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 - 0.623T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + 1.04T + 89T^{2} \) |
| 97 | \( 1 + 9.78T + 97T^{2} \) |
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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
−9.549338018690553161237485285697, −8.686859641166875215303535875120, −7.955093476335300765316419684753, −7.49658144780132882148536001299, −6.39172316760615194863125916091, −5.20370386624825582440955764997, −4.27156307730682706183839017510, −3.77283853050391345538002133523, −2.70510539178893029447659915696, −0.19291232981987406185582895906,
0.19291232981987406185582895906, 2.70510539178893029447659915696, 3.77283853050391345538002133523, 4.27156307730682706183839017510, 5.20370386624825582440955764997, 6.39172316760615194863125916091, 7.49658144780132882148536001299, 7.955093476335300765316419684753, 8.686859641166875215303535875120, 9.549338018690553161237485285697
