L(s) = 1 | + 5.30·2-s − 3·3-s + 20.1·4-s + 5.56·5-s − 15.9·6-s + 64.6·8-s + 9·9-s + 29.5·10-s − 13.9·11-s − 60.5·12-s + 38.6·13-s − 16.6·15-s + 181.·16-s + 43.4·17-s + 47.7·18-s − 109.·19-s + 112.·20-s − 73.8·22-s − 74.8·23-s − 193.·24-s − 94.0·25-s + 205.·26-s − 27·27-s − 72.3·29-s − 88.5·30-s + 64.0·31-s + 447.·32-s + ⋯ |
L(s) = 1 | + 1.87·2-s − 0.577·3-s + 2.52·4-s + 0.497·5-s − 1.08·6-s + 2.85·8-s + 0.333·9-s + 0.933·10-s − 0.381·11-s − 1.45·12-s + 0.825·13-s − 0.287·15-s + 2.83·16-s + 0.620·17-s + 0.625·18-s − 1.31·19-s + 1.25·20-s − 0.715·22-s − 0.678·23-s − 1.64·24-s − 0.752·25-s + 1.54·26-s − 0.192·27-s − 0.463·29-s − 0.538·30-s + 0.371·31-s + 2.47·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.485617879\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.485617879\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.30T + 8T^{2} \) |
| 5 | \( 1 - 5.56T + 125T^{2} \) |
| 11 | \( 1 + 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 383.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 79.2T + 9.12e5T^{2} \) |
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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
−12.85312649934893993199472607351, −11.81377075701963421647725071493, −11.00163113781393471238130629589, −10.02062867986442870807431738009, −7.970124964292707008190714799467, −6.49972481061970819624733738703, −5.89786412321465114087629484668, −4.80459895116996090998856457795, −3.60029489466847662311093299728, −1.96703866863229147895545859262,
1.96703866863229147895545859262, 3.60029489466847662311093299728, 4.80459895116996090998856457795, 5.89786412321465114087629484668, 6.49972481061970819624733738703, 7.970124964292707008190714799467, 10.02062867986442870807431738009, 11.00163113781393471238130629589, 11.81377075701963421647725071493, 12.85312649934893993199472607351