| L(s) = 1 | + 2-s − 0.0462·3-s + 4-s + 2.81·5-s − 0.0462·6-s − 4.11·7-s + 8-s − 2.99·9-s + 2.81·10-s + 5.30·11-s − 0.0462·12-s − 4.11·14-s − 0.130·15-s + 16-s − 2.26·17-s − 2.99·18-s + 19-s + 2.81·20-s + 0.190·21-s + 5.30·22-s − 2.21·23-s − 0.0462·24-s + 2.94·25-s + 0.277·27-s − 4.11·28-s − 6.69·29-s − 0.130·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0267·3-s + 0.5·4-s + 1.26·5-s − 0.0188·6-s − 1.55·7-s + 0.353·8-s − 0.999·9-s + 0.891·10-s + 1.60·11-s − 0.0133·12-s − 1.10·14-s − 0.0336·15-s + 0.250·16-s − 0.548·17-s − 0.706·18-s + 0.229·19-s + 0.630·20-s + 0.0415·21-s + 1.13·22-s − 0.462·23-s − 0.00944·24-s + 0.588·25-s + 0.0534·27-s − 0.778·28-s − 1.24·29-s − 0.0238·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411732613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411732613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0462T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 23 | \( 1 + 2.21T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 5.50T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.36T + 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 8.52T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.19T + 97T^{2} \) |
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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
−7.941288540827067106414231600453, −6.89729266675732123282450067372, −6.35571663083092676979323576056, −5.99467924745811571622304361551, −5.47351582379989614409573197502, −4.24207667094688781272488542169, −3.65890127618121867886108082729, −2.73293380923226548192313510320, −2.16589498899022442133958509025, −0.852680760968564892575035601725,
0.852680760968564892575035601725, 2.16589498899022442133958509025, 2.73293380923226548192313510320, 3.65890127618121867886108082729, 4.24207667094688781272488542169, 5.47351582379989614409573197502, 5.99467924745811571622304361551, 6.35571663083092676979323576056, 6.89729266675732123282450067372, 7.941288540827067106414231600453
