| L(s) = 1 | − 1.35·2-s + 2.90·3-s − 0.171·4-s − 2.71·5-s − 3.92·6-s − 3.82·7-s + 2.93·8-s + 5.43·9-s + 3.66·10-s − 11-s − 0.497·12-s + 6.55·13-s + 5.17·14-s − 7.87·15-s − 3.62·16-s − 17-s − 7.34·18-s − 4.53·19-s + 0.465·20-s − 11.1·21-s + 1.35·22-s − 5.04·23-s + 8.52·24-s + 2.35·25-s − 8.86·26-s + 7.05·27-s + 0.655·28-s + ⋯ |
| L(s) = 1 | − 0.956·2-s + 1.67·3-s − 0.0857·4-s − 1.21·5-s − 1.60·6-s − 1.44·7-s + 1.03·8-s + 1.81·9-s + 1.15·10-s − 0.301·11-s − 0.143·12-s + 1.81·13-s + 1.38·14-s − 2.03·15-s − 0.906·16-s − 0.242·17-s − 1.73·18-s − 1.04·19-s + 0.103·20-s − 2.42·21-s + 0.288·22-s − 1.05·23-s + 1.74·24-s + 0.471·25-s − 1.73·26-s + 1.35·27-s + 0.123·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019116137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019116137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 13 | \( 1 - 6.55T + 13T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 - 1.78T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 47 | \( 1 - 2.93T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 7.03T + 67T^{2} \) |
| 71 | \( 1 - 9.18T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 5.82T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.002225432872140133219774040489, −7.61168433287161218701485858772, −6.70144065371227141336617386761, −6.14633032489779072175891986250, −4.59533493166905722538168427700, −3.82968804260071335470357856291, −3.64618812533234321230525618704, −2.73979033347501959332385399915, −1.73313804326769311636259902425, −0.52973734165371518273361532254,
0.52973734165371518273361532254, 1.73313804326769311636259902425, 2.73979033347501959332385399915, 3.64618812533234321230525618704, 3.82968804260071335470357856291, 4.59533493166905722538168427700, 6.14633032489779072175891986250, 6.70144065371227141336617386761, 7.61168433287161218701485858772, 8.002225432872140133219774040489
