| L(s) = 1 | + 1.46·2-s − 83.8·3-s − 509.·4-s − 1.97e3·5-s − 122.·6-s − 7.40e3·7-s − 1.49e3·8-s − 1.26e4·9-s − 2.89e3·10-s + 5.60e3·11-s + 4.27e4·12-s − 8.11e4·13-s − 1.08e4·14-s + 1.65e5·15-s + 2.58e5·16-s − 5.20e4·17-s − 1.85e4·18-s − 2.43e5·19-s + 1.00e6·20-s + 6.20e5·21-s + 8.20e3·22-s − 2.29e6·23-s + 1.25e5·24-s + 1.95e6·25-s − 1.18e5·26-s + 2.71e6·27-s + 3.77e6·28-s + ⋯ |
| L(s) = 1 | + 0.0646·2-s − 0.597·3-s − 0.995·4-s − 1.41·5-s − 0.0386·6-s − 1.16·7-s − 0.129·8-s − 0.642·9-s − 0.0915·10-s + 0.115·11-s + 0.595·12-s − 0.787·13-s − 0.0753·14-s + 0.845·15-s + 0.987·16-s − 0.151·17-s − 0.0415·18-s − 0.429·19-s + 1.40·20-s + 0.696·21-s + 0.00746·22-s − 1.71·23-s + 0.0771·24-s + 1.00·25-s − 0.0509·26-s + 0.981·27-s + 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.01379288624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01379288624\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 1.46T + 512T^{2} \) |
| 3 | \( 1 + 83.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.97e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.60e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.11e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.20e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.29e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.00e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.10e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.01e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 5.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.91e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.00e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.67e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.22e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.44e8T + 7.60e17T^{2} \) |
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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
−13.94021980324412002533158843542, −12.51633929971384325673255020444, −11.89303826397020972770399292879, −10.36565556130127164876150433017, −9.011424811491467973225463896367, −7.74473833545449555597152144491, −6.10495998194128047382633802076, −4.53744405280158807001936325626, −3.34235794869908244047910660464, −0.07612925234404484027660829254,
0.07612925234404484027660829254, 3.34235794869908244047910660464, 4.53744405280158807001936325626, 6.10495998194128047382633802076, 7.74473833545449555597152144491, 9.011424811491467973225463896367, 10.36565556130127164876150433017, 11.89303826397020972770399292879, 12.51633929971384325673255020444, 13.94021980324412002533158843542
