L(s) = 1 | − 1.92·2-s − 3.23·3-s + 1.71·4-s + 1.34·5-s + 6.24·6-s + 3.96·7-s + 0.556·8-s + 7.49·9-s − 2.60·10-s − 2.26·11-s − 5.54·12-s − 1.74·13-s − 7.64·14-s − 4.37·15-s − 4.49·16-s + 2.07·17-s − 14.4·18-s − 0.0556·19-s + 2.31·20-s − 12.8·21-s + 4.36·22-s + 2.13·23-s − 1.80·24-s − 3.17·25-s + 3.35·26-s − 14.5·27-s + 6.78·28-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 1.87·3-s + 0.855·4-s + 0.603·5-s + 2.54·6-s + 1.49·7-s + 0.196·8-s + 2.49·9-s − 0.822·10-s − 0.683·11-s − 1.60·12-s − 0.482·13-s − 2.04·14-s − 1.12·15-s − 1.12·16-s + 0.503·17-s − 3.40·18-s − 0.0127·19-s + 0.516·20-s − 2.80·21-s + 0.930·22-s + 0.445·23-s − 0.367·24-s − 0.635·25-s + 0.657·26-s − 2.80·27-s + 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5929282405\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5929282405\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.0556T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 41 | \( 1 - 9.82T + 41T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 - 0.851T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 + 0.341T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.0238T + 97T^{2} \) |
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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
−8.308497091617948106913281692941, −7.69147218800972141047390738738, −7.19417376223587981313749719661, −6.22386942920272563068889560916, −5.52778108614865712988069088534, −4.88089712274455431853773961229, −4.33822767612882821154539931011, −2.30132371683464671018428362725, −1.44011090126569029944452394257, −0.66222133233296294196973832381,
0.66222133233296294196973832381, 1.44011090126569029944452394257, 2.30132371683464671018428362725, 4.33822767612882821154539931011, 4.88089712274455431853773961229, 5.52778108614865712988069088534, 6.22386942920272563068889560916, 7.19417376223587981313749719661, 7.69147218800972141047390738738, 8.308497091617948106913281692941