L(s) = 1 | + 2.80·2-s + 3-s + 5.86·4-s + 2.52·5-s + 2.80·6-s − 2.67·7-s + 10.8·8-s + 9-s + 7.08·10-s − 0.714·11-s + 5.86·12-s + 0.530·13-s − 7.48·14-s + 2.52·15-s + 18.6·16-s − 8.04·17-s + 2.80·18-s − 5.87·19-s + 14.8·20-s − 2.67·21-s − 2.00·22-s + 23-s + 10.8·24-s + 1.39·25-s + 1.48·26-s + 27-s − 15.6·28-s + ⋯ |
L(s) = 1 | + 1.98·2-s + 0.577·3-s + 2.93·4-s + 1.13·5-s + 1.14·6-s − 1.00·7-s + 3.82·8-s + 0.333·9-s + 2.24·10-s − 0.215·11-s + 1.69·12-s + 0.147·13-s − 2.00·14-s + 0.652·15-s + 4.65·16-s − 1.95·17-s + 0.660·18-s − 1.34·19-s + 3.31·20-s − 0.582·21-s − 0.427·22-s + 0.208·23-s + 2.20·24-s + 0.278·25-s + 0.291·26-s + 0.192·27-s − 2.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.175169398\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.175169398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 + 0.714T + 11T^{2} \) |
| 13 | \( 1 - 0.530T + 13T^{2} \) |
| 17 | \( 1 + 8.04T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 31 | \( 1 + 0.921T + 31T^{2} \) |
| 37 | \( 1 + 0.783T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6.08T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 - 1.50T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.258357783584941017057454953631, −8.253314820341513253130916578445, −7.01467386370729644928378148530, −6.48980679568714908395261038649, −6.04919643680700665266627165706, −5.00948745238288672948733090248, −4.24804403440441967587833530258, −3.37093051809576792999547809720, −2.42275656485589492409412113831, −1.95396464788166910518470597010,
1.95396464788166910518470597010, 2.42275656485589492409412113831, 3.37093051809576792999547809720, 4.24804403440441967587833530258, 5.00948745238288672948733090248, 6.04919643680700665266627165706, 6.48980679568714908395261038649, 7.01467386370729644928378148530, 8.253314820341513253130916578445, 9.258357783584941017057454953631