L(s) = 1 | − 2.40·2-s − 1.47·3-s + 3.78·4-s − 0.987·5-s + 3.54·6-s + 2.06·7-s − 4.29·8-s − 0.827·9-s + 2.37·10-s + 11-s − 5.57·12-s − 2.64·13-s − 4.95·14-s + 1.45·15-s + 2.76·16-s + 17-s + 1.99·18-s + 3.84·19-s − 3.73·20-s − 3.03·21-s − 2.40·22-s − 3.24·23-s + 6.33·24-s − 4.02·25-s + 6.36·26-s + 5.64·27-s + 7.80·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 0.850·3-s + 1.89·4-s − 0.441·5-s + 1.44·6-s + 0.779·7-s − 1.51·8-s − 0.275·9-s + 0.750·10-s + 0.301·11-s − 1.61·12-s − 0.733·13-s − 1.32·14-s + 0.375·15-s + 0.690·16-s + 0.242·17-s + 0.469·18-s + 0.883·19-s − 0.835·20-s − 0.663·21-s − 0.512·22-s − 0.677·23-s + 1.29·24-s − 0.805·25-s + 1.24·26-s + 1.08·27-s + 1.47·28-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
\(L(1)\) |
\(\approx\) |
\(0.4082714984\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4082714984\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 + 0.987T + 5T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 8.79T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 53 | \( 1 + 5.18T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 8.60T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 6.59T + 97T^{2} \) |
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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
−7.75986675024691236663910398273, −7.53104494133193822856908730095, −6.61086422796577373364498775481, −6.00397634729185596517381081548, −5.14174418000648198065997991930, −4.46879901866127170656731257680, −3.26334783554716080247294489864, −2.27626189337143221807779476153, −1.36305990535923110158022186759, −0.46304338539659712584564274333,
0.46304338539659712584564274333, 1.36305990535923110158022186759, 2.27626189337143221807779476153, 3.26334783554716080247294489864, 4.46879901866127170656731257680, 5.14174418000648198065997991930, 6.00397634729185596517381081548, 6.61086422796577373364498775481, 7.53104494133193822856908730095, 7.75986675024691236663910398273