| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 17.6·5-s − 6·6-s + 14·7-s − 8·8-s + 9·9-s − 35.2·10-s − 17.6·11-s + 12·12-s − 9.28·13-s − 28·14-s + 52.9·15-s + 16·16-s − 35.2·17-s − 18·18-s + 55.6·19-s + 70.5·20-s + 42·21-s + 35.2·22-s + 23·23-s − 24·24-s + 186.·25-s + 18.5·26-s + 27·27-s + 56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.57·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s − 1.11·10-s − 0.483·11-s + 0.288·12-s − 0.198·13-s − 0.534·14-s + 0.911·15-s + 0.250·16-s − 0.503·17-s − 0.235·18-s + 0.671·19-s + 0.789·20-s + 0.436·21-s + 0.341·22-s + 0.208·23-s − 0.204·24-s + 1.49·25-s + 0.140·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.953478755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953478755\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 17.6T + 125T^{2} \) |
| 7 | \( 1 - 14T + 343T^{2} \) |
| 11 | \( 1 + 17.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.28T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 74.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 64.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 379.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 396T + 2.05e5T^{2} \) |
| 61 | \( 1 - 594.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 911.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 941.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 520.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 418.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 599.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 40.7T + 9.12e5T^{2} \) |
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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
−12.92547374292640140061553706827, −11.46346895271236946855007320141, −10.32544424462697176954639539465, −9.599129657065987123088264412173, −8.663072910594084586633661286783, −7.56443054271149334091599031904, −6.24402351624511836170189210087, −4.97182291819632518984955449146, −2.69011601631536817334069759172, −1.54757530377360437695308602992,
1.54757530377360437695308602992, 2.69011601631536817334069759172, 4.97182291819632518984955449146, 6.24402351624511836170189210087, 7.56443054271149334091599031904, 8.663072910594084586633661286783, 9.599129657065987123088264412173, 10.32544424462697176954639539465, 11.46346895271236946855007320141, 12.92547374292640140061553706827
