| L(s) = 1 | − 150.·2-s + 1.46e4·4-s + 1.56e4·5-s − 2.42e5·7-s − 9.68e5·8-s − 2.35e6·10-s + 6.86e6·11-s + 2.01e7·13-s + 3.66e7·14-s + 2.65e7·16-s + 1.41e8·17-s − 2.08e8·19-s + 2.28e8·20-s − 1.03e9·22-s + 2.35e8·23-s + 2.44e8·25-s − 3.03e9·26-s − 3.54e9·28-s − 2.00e9·29-s − 3.36e9·31-s + 3.92e9·32-s − 2.13e10·34-s − 3.79e9·35-s − 9.62e9·37-s + 3.14e10·38-s − 1.51e10·40-s − 2.34e10·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.78·4-s + 0.447·5-s − 0.780·7-s − 1.30·8-s − 0.746·10-s + 1.16·11-s + 1.15·13-s + 1.30·14-s + 0.396·16-s + 1.41·17-s − 1.01·19-s + 0.797·20-s − 1.94·22-s + 0.331·23-s + 0.199·25-s − 1.92·26-s − 1.39·28-s − 0.627·29-s − 0.680·31-s + 0.645·32-s − 2.36·34-s − 0.348·35-s − 0.616·37-s + 1.69·38-s − 0.584·40-s − 0.769·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9888364241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9888364241\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 1.56e4T \) |
| good | 2 | \( 1 + 150.T + 8.19e3T^{2} \) |
| 7 | \( 1 + 2.42e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.86e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.01e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.41e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.08e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.35e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.00e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.36e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 9.62e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.34e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 8.82e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.04e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.44e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 1.25e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 7.31e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.13e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 2.11e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.39e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.88e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 6.14e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.90e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.60e13T + 6.73e25T^{2} \) |
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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
−12.78836061043571676960818670360, −11.37876628629082936205378061896, −10.27040479354659450103654061359, −9.345297371713592833352759559289, −8.487321401328237767794876212722, −7.00896684629682708951516429900, −6.02137491401017631859419212708, −3.53407993869415968887218564115, −1.79323851518260483269720167072, −0.75244812591360242009890165149,
0.75244812591360242009890165149, 1.79323851518260483269720167072, 3.53407993869415968887218564115, 6.02137491401017631859419212708, 7.00896684629682708951516429900, 8.487321401328237767794876212722, 9.345297371713592833352759559289, 10.27040479354659450103654061359, 11.37876628629082936205378061896, 12.78836061043571676960818670360
