| L(s) = 1 | − 2.59·2-s + 254.·3-s − 505.·4-s − 1.90e3·5-s − 658.·6-s + 2.40e3·7-s + 2.63e3·8-s + 4.49e4·9-s + 4.93e3·10-s + 1.83e4·11-s − 1.28e5·12-s − 2.85e4·13-s − 6.22e3·14-s − 4.84e5·15-s + 2.51e5·16-s − 3.73e5·17-s − 1.16e5·18-s − 2.46e5·19-s + 9.61e5·20-s + 6.10e5·21-s − 4.75e4·22-s + 1.88e6·23-s + 6.70e5·24-s + 1.67e6·25-s + 7.40e4·26-s + 6.42e6·27-s − 1.21e6·28-s + ⋯ |
| L(s) = 1 | − 0.114·2-s + 1.81·3-s − 0.986·4-s − 1.36·5-s − 0.207·6-s + 0.377·7-s + 0.227·8-s + 2.28·9-s + 0.156·10-s + 0.377·11-s − 1.78·12-s − 0.277·13-s − 0.0432·14-s − 2.46·15-s + 0.960·16-s − 1.08·17-s − 0.261·18-s − 0.433·19-s + 1.34·20-s + 0.685·21-s − 0.0432·22-s + 1.40·23-s + 0.412·24-s + 0.855·25-s + 0.0317·26-s + 2.32·27-s − 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.483203161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483203161\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 2.59T + 512T^{2} \) |
| 3 | \( 1 - 254.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.90e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.83e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.73e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.88e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.80e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.76e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 7.15e4T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.72e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.71e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.56e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.06e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 9.59e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.19e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.56e8T + 7.60e17T^{2} \) |
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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
−12.62531847725147270806061320711, −11.12033009974611848381617789232, −9.598629658791225915887161883305, −8.740039832174320427770483459566, −8.108792730532787776362402712374, −7.18704709263621692882903710679, −4.49995570158992380618440400237, −3.95568770919999740026526756632, −2.64620203620183362337787860214, −0.871451186995177830713079223319,
0.871451186995177830713079223319, 2.64620203620183362337787860214, 3.95568770919999740026526756632, 4.49995570158992380618440400237, 7.18704709263621692882903710679, 8.108792730532787776362402712374, 8.740039832174320427770483459566, 9.598629658791225915887161883305, 11.12033009974611848381617789232, 12.62531847725147270806061320711
