| L(s) = 1 | − 4.08·2-s − 9.96·3-s + 8.67·4-s + 12.0·5-s + 40.7·6-s + 17.5·7-s − 2.76·8-s + 72.3·9-s − 49.3·10-s + 10.8·11-s − 86.4·12-s − 42.2·13-s − 71.5·14-s − 120.·15-s − 58.1·16-s − 5.55·17-s − 295.·18-s + 154.·19-s + 104.·20-s − 174.·21-s − 44.3·22-s − 113.·23-s + 27.6·24-s + 20.7·25-s + 172.·26-s − 451.·27-s + 152.·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 1.91·3-s + 1.08·4-s + 1.07·5-s + 2.76·6-s + 0.946·7-s − 0.122·8-s + 2.67·9-s − 1.55·10-s + 0.297·11-s − 2.08·12-s − 0.901·13-s − 1.36·14-s − 2.07·15-s − 0.908·16-s − 0.0793·17-s − 3.86·18-s + 1.87·19-s + 1.17·20-s − 1.81·21-s − 0.429·22-s − 1.02·23-s + 0.234·24-s + 0.166·25-s + 1.30·26-s − 3.21·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7197189712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7197189712\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 4.08T + 8T^{2} \) |
| 3 | \( 1 + 9.96T + 27T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 - 17.5T + 343T^{2} \) |
| 11 | \( 1 - 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.55T + 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 23.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 35.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 285.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 93.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 199.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 622.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 687.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 128.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 353.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 518.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 616.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 92.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−9.284852543516561812879882082871, −7.932903016392395051057720489356, −7.37462665636235578378618762830, −6.58900782141764341206554396135, −5.70586621706175160995307277176, −5.15400328371339380557595257172, −4.30319133714099629386161803037, −2.15565551005936847848857810899, −1.36533761958860830951600423638, −0.62082835630608501293579384900,
0.62082835630608501293579384900, 1.36533761958860830951600423638, 2.15565551005936847848857810899, 4.30319133714099629386161803037, 5.15400328371339380557595257172, 5.70586621706175160995307277176, 6.58900782141764341206554396135, 7.37462665636235578378618762830, 7.932903016392395051057720489356, 9.284852543516561812879882082871
