L(s) = 1 | + 1.32·2-s + 3-s − 0.239·4-s + 0.326·5-s + 1.32·6-s − 7-s − 2.97·8-s + 9-s + 0.433·10-s − 0.239·12-s − 0.0589·13-s − 1.32·14-s + 0.326·15-s − 3.46·16-s + 6.28·17-s + 1.32·18-s + 2.43·19-s − 0.0782·20-s − 21-s + 5.93·23-s − 2.97·24-s − 4.89·25-s − 0.0782·26-s + 27-s + 0.239·28-s + 9.43·29-s + 0.433·30-s + ⋯ |
L(s) = 1 | + 0.938·2-s + 0.577·3-s − 0.119·4-s + 0.146·5-s + 0.541·6-s − 0.377·7-s − 1.05·8-s + 0.333·9-s + 0.137·10-s − 0.0690·12-s − 0.0163·13-s − 0.354·14-s + 0.0844·15-s − 0.866·16-s + 1.52·17-s + 0.312·18-s + 0.558·19-s − 0.0174·20-s − 0.218·21-s + 1.23·23-s − 0.606·24-s − 0.978·25-s − 0.0153·26-s + 0.192·27-s + 0.0452·28-s + 1.75·29-s + 0.0791·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.225990761\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.225990761\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 - 0.326T + 5T^{2} \) |
| 13 | \( 1 + 0.0589T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 - 1.98T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 + 0.0782T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−8.873098696257806529625207598281, −8.210693071958330554767265892506, −7.29472318339927871129214899884, −6.47410961223211854268125980772, −5.57857385569260262819008063124, −4.99305096851432735166275557852, −3.98429375328613011950614477817, −3.27696248668565379865974303419, −2.60073375527448046980541230878, −1.00541820926269119317767085626,
1.00541820926269119317767085626, 2.60073375527448046980541230878, 3.27696248668565379865974303419, 3.98429375328613011950614477817, 4.99305096851432735166275557852, 5.57857385569260262819008063124, 6.47410961223211854268125980772, 7.29472318339927871129214899884, 8.210693071958330554767265892506, 8.873098696257806529625207598281