| L(s) = 1 | + 0.381·3-s + 2.23·5-s − 4.61·7-s − 2.85·9-s − 4·11-s + 5.61·13-s + 0.854·15-s + 3.09·17-s + 0.472·19-s − 1.76·21-s − 5.85·23-s − 2.23·27-s − 6.23·29-s + 0.763·31-s − 1.52·33-s − 10.3·35-s − 7.94·37-s + 2.14·39-s − 2.23·41-s − 7.47·43-s − 6.38·45-s + 0.236·47-s + 14.3·49-s + 1.18·51-s − 9.23·53-s − 8.94·55-s + 0.180·57-s + ⋯ |
| L(s) = 1 | + 0.220·3-s + 0.999·5-s − 1.74·7-s − 0.951·9-s − 1.20·11-s + 1.55·13-s + 0.220·15-s + 0.749·17-s + 0.108·19-s − 0.384·21-s − 1.22·23-s − 0.430·27-s − 1.15·29-s + 0.137·31-s − 0.265·33-s − 1.74·35-s − 1.30·37-s + 0.343·39-s − 0.349·41-s − 1.13·43-s − 0.951·45-s + 0.0344·47-s + 2.04·49-s + 0.165·51-s − 1.26·53-s − 1.20·55-s + 0.0238·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 - 0.472T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 + 7.94T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 0.236T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−9.302184820149150234286194945572, −8.529785474698456320952575867035, −7.67901585160023523786338651621, −6.42041195525743179557296102796, −5.95888083018808687972341007117, −5.35599807592408579306289928041, −3.59038851948870106207030678221, −3.10730484236341689964947889114, −1.93164184302037806970087415118, 0,
1.93164184302037806970087415118, 3.10730484236341689964947889114, 3.59038851948870106207030678221, 5.35599807592408579306289928041, 5.95888083018808687972341007117, 6.42041195525743179557296102796, 7.67901585160023523786338651621, 8.529785474698456320952575867035, 9.302184820149150234286194945572
