L(s) = 1 | − 2.49·2-s + 1.26·3-s + 4.20·4-s − 3.18·5-s − 3.14·6-s + 4.16·7-s − 5.50·8-s − 1.40·9-s + 7.94·10-s + 4.10·11-s + 5.32·12-s − 2.81·13-s − 10.3·14-s − 4.02·15-s + 5.29·16-s − 5.22·17-s + 3.49·18-s + 4.32·19-s − 13.4·20-s + 5.26·21-s − 10.2·22-s − 2.96·23-s − 6.95·24-s + 5.16·25-s + 7.00·26-s − 5.56·27-s + 17.5·28-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.729·3-s + 2.10·4-s − 1.42·5-s − 1.28·6-s + 1.57·7-s − 1.94·8-s − 0.467·9-s + 2.51·10-s + 1.23·11-s + 1.53·12-s − 0.779·13-s − 2.77·14-s − 1.04·15-s + 1.32·16-s − 1.26·17-s + 0.823·18-s + 0.992·19-s − 2.99·20-s + 1.14·21-s − 2.18·22-s − 0.618·23-s − 1.42·24-s + 1.03·25-s + 1.37·26-s − 1.07·27-s + 3.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7731101244\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7731101244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 - 0.134T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 0.709T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2.60T + 61T^{2} \) |
| 67 | \( 1 - 7.18T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 6.14T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 + 0.627T + 89T^{2} \) |
| 97 | \( 1 + 9.52T + 97T^{2} \) |
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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
−7.995811862992686932562520990887, −7.45205821110221613308491192253, −7.10311637168180148701967113099, −6.06608020089268705910512193818, −4.87362621518714193663719799398, −4.16844578827602368770612741830, −3.31589350411568382173248957749, −2.28467774100128796520019714422, −1.64815124112360076206230719586, −0.55280200247481307597420827221,
0.55280200247481307597420827221, 1.64815124112360076206230719586, 2.28467774100128796520019714422, 3.31589350411568382173248957749, 4.16844578827602368770612741830, 4.87362621518714193663719799398, 6.06608020089268705910512193818, 7.10311637168180148701967113099, 7.45205821110221613308491192253, 7.995811862992686932562520990887