| L(s) = 1 | + 1.39·2-s + 1.35·3-s − 0.0576·4-s − 3.24·5-s + 1.89·6-s + 2.34·7-s − 2.86·8-s − 1.15·9-s − 4.52·10-s − 11-s − 0.0781·12-s + 1.47·13-s + 3.26·14-s − 4.40·15-s − 3.88·16-s − 1.07·17-s − 1.61·18-s − 2.18·19-s + 0.186·20-s + 3.18·21-s − 1.39·22-s − 7.43·23-s − 3.89·24-s + 5.52·25-s + 2.05·26-s − 5.64·27-s − 0.135·28-s + ⋯ |
| L(s) = 1 | + 0.985·2-s + 0.783·3-s − 0.0288·4-s − 1.45·5-s + 0.772·6-s + 0.886·7-s − 1.01·8-s − 0.385·9-s − 1.42·10-s − 0.301·11-s − 0.0225·12-s + 0.409·13-s + 0.873·14-s − 1.13·15-s − 0.970·16-s − 0.260·17-s − 0.380·18-s − 0.500·19-s + 0.0417·20-s + 0.694·21-s − 0.297·22-s − 1.55·23-s − 0.794·24-s + 1.10·25-s + 0.403·26-s − 1.08·27-s − 0.0255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 - 1.35T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 + 0.595T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 - 4.36T + 59T^{2} \) |
| 61 | \( 1 + 2.12T + 61T^{2} \) |
| 67 | \( 1 + 0.285T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 - 1.08T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.749426273342204398128293969214, −8.251743077382310461378959855568, −7.83249023551762908958267878651, −6.62077663564792479968050364626, −5.55371461438151769154034340105, −4.66603621495671929834922359983, −3.90714722044542811007492219727, −3.35779364917165037140613310786, −2.16009041612731150915289129956, 0,
2.16009041612731150915289129956, 3.35779364917165037140613310786, 3.90714722044542811007492219727, 4.66603621495671929834922359983, 5.55371461438151769154034340105, 6.62077663564792479968050364626, 7.83249023551762908958267878651, 8.251743077382310461378959855568, 8.749426273342204398128293969214
