| L(s) = 1 | + 4.00·2-s + 8.04·4-s − 3.38·5-s + 23.4·7-s + 0.172·8-s − 13.5·10-s + 59.8·11-s − 84.9·13-s + 94.0·14-s − 63.6·16-s − 78.1·17-s + 99.6·19-s − 27.2·20-s + 239.·22-s − 172.·23-s − 113.·25-s − 340.·26-s + 188.·28-s − 127.·29-s − 99.5·31-s − 256.·32-s − 312.·34-s − 79.4·35-s − 127.·37-s + 399.·38-s − 0.584·40-s − 497.·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.00·4-s − 0.302·5-s + 1.26·7-s + 0.00763·8-s − 0.428·10-s + 1.64·11-s − 1.81·13-s + 1.79·14-s − 0.994·16-s − 1.11·17-s + 1.20·19-s − 0.304·20-s + 2.32·22-s − 1.56·23-s − 0.908·25-s − 2.56·26-s + 1.27·28-s − 0.818·29-s − 0.576·31-s − 1.41·32-s − 1.57·34-s − 0.383·35-s − 0.567·37-s + 1.70·38-s − 0.00230·40-s − 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 4.00T + 8T^{2} \) |
| 5 | \( 1 + 3.38T + 125T^{2} \) |
| 7 | \( 1 - 23.4T + 343T^{2} \) |
| 11 | \( 1 - 59.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 497.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 447.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 141.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 303.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 670.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 836.T + 9.12e5T^{2} \) |
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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
−8.177259007110122981381653942176, −7.30547790493616856530359123185, −6.73035297061375180653448204492, −5.60872233092329463290624793093, −5.08443995307016238677065630359, −4.13972407221446324138786919108, −3.85941228410604550471490896440, −2.43343852587956473787085485353, −1.69200815905597853022829689032, 0,
1.69200815905597853022829689032, 2.43343852587956473787085485353, 3.85941228410604550471490896440, 4.13972407221446324138786919108, 5.08443995307016238677065630359, 5.60872233092329463290624793093, 6.73035297061375180653448204492, 7.30547790493616856530359123185, 8.177259007110122981381653942176
