L(s) = 1 | + 2.26·2-s − 3-s + 3.11·4-s − 3.51·5-s − 2.26·6-s + 2.53·8-s + 9-s − 7.95·10-s − 3.79·11-s − 3.11·12-s − 1.46·13-s + 3.51·15-s − 0.508·16-s − 6.67·17-s + 2.26·18-s + 2.27·19-s − 10.9·20-s − 8.58·22-s + 8.49·23-s − 2.53·24-s + 7.35·25-s − 3.30·26-s − 27-s − 5.04·29-s + 7.95·30-s + 1.44·31-s − 6.21·32-s + ⋯ |
L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.55·4-s − 1.57·5-s − 0.923·6-s + 0.895·8-s + 0.333·9-s − 2.51·10-s − 1.14·11-s − 0.900·12-s − 0.405·13-s + 0.907·15-s − 0.127·16-s − 1.61·17-s + 0.533·18-s + 0.522·19-s − 2.45·20-s − 1.82·22-s + 1.77·23-s − 0.516·24-s + 1.47·25-s − 0.648·26-s − 0.192·27-s − 0.937·29-s + 1.45·30-s + 0.260·31-s − 1.09·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.953534325\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.953534325\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 + 5.04T + 29T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 0.996T + 37T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.748T + 47T^{2} \) |
| 53 | \( 1 - 9.47T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 1.76T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.74559559791741606280925352740, −6.98644854085745216001966986283, −6.82768535240890506682507548351, −5.44144032542335490038957582485, −5.27230589977359215410068593347, −4.30979902834890490333847164565, −4.02696078951280988433964098345, −2.99833568438532750696940782916, −2.36708065177350825287052106744, −0.56018605837779275550775265647,
0.56018605837779275550775265647, 2.36708065177350825287052106744, 2.99833568438532750696940782916, 4.02696078951280988433964098345, 4.30979902834890490333847164565, 5.27230589977359215410068593347, 5.44144032542335490038957582485, 6.82768535240890506682507548351, 6.98644854085745216001966986283, 7.74559559791741606280925352740