| L(s) = 1 | + 2-s − 3-s + 4-s − 0.120·5-s − 6-s − 4.29·7-s + 8-s + 9-s − 0.120·10-s + 2.57·11-s − 12-s + 0.815·13-s − 4.29·14-s + 0.120·15-s + 16-s − 0.467·17-s + 18-s − 0.120·20-s + 4.29·21-s + 2.57·22-s + 5.55·23-s − 24-s − 4.98·25-s + 0.815·26-s − 27-s − 4.29·28-s + 2.34·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0539·5-s − 0.408·6-s − 1.62·7-s + 0.353·8-s + 0.333·9-s − 0.0381·10-s + 0.776·11-s − 0.288·12-s + 0.226·13-s − 1.14·14-s + 0.0311·15-s + 0.250·16-s − 0.113·17-s + 0.235·18-s − 0.0269·20-s + 0.936·21-s + 0.548·22-s + 1.15·23-s − 0.204·24-s − 0.997·25-s + 0.159·26-s − 0.192·27-s − 0.810·28-s + 0.435·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.916906043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916906043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 0.120T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.815T + 13T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 - 5.34T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 - 3.10T + 97T^{2} \) |
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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
−9.277314184523635179495568983900, −8.272652510254526832897274600048, −6.97860466397803875160037265232, −6.72055175034900457034171013075, −5.95223525895668392270859585770, −5.18542897435060327636616578344, −4.05489412568529068171761198682, −3.48776306539571719315407114457, −2.42122008440997029676632606496, −0.839624117923517940111770228900,
0.839624117923517940111770228900, 2.42122008440997029676632606496, 3.48776306539571719315407114457, 4.05489412568529068171761198682, 5.18542897435060327636616578344, 5.95223525895668392270859585770, 6.72055175034900457034171013075, 6.97860466397803875160037265232, 8.272652510254526832897274600048, 9.277314184523635179495568983900
