L(s) = 1 | + 5.33·2-s + 6.59·3-s + 20.5·4-s − 12.0·5-s + 35.2·6-s − 18.2·7-s + 66.8·8-s + 16.5·9-s − 64.4·10-s − 44.5·11-s + 135.·12-s − 45.4·13-s − 97.3·14-s − 79.6·15-s + 192.·16-s − 98.4·17-s + 88.1·18-s − 86.0·19-s − 247.·20-s − 120.·21-s − 237.·22-s + 142.·23-s + 440.·24-s + 20.8·25-s − 242.·26-s − 69.2·27-s − 373.·28-s + ⋯ |
L(s) = 1 | + 1.88·2-s + 1.26·3-s + 2.56·4-s − 1.08·5-s + 2.39·6-s − 0.984·7-s + 2.95·8-s + 0.611·9-s − 2.03·10-s − 1.22·11-s + 3.25·12-s − 0.969·13-s − 1.85·14-s − 1.37·15-s + 3.01·16-s − 1.40·17-s + 1.15·18-s − 1.03·19-s − 2.76·20-s − 1.24·21-s − 2.30·22-s + 1.29·23-s + 3.74·24-s + 0.166·25-s − 1.83·26-s − 0.493·27-s − 2.52·28-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 5.33T + 8T^{2} \) |
| 3 | \( 1 - 6.59T + 27T^{2} \) |
| 5 | \( 1 + 12.0T + 125T^{2} \) |
| 7 | \( 1 + 18.2T + 343T^{2} \) |
| 11 | \( 1 + 44.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 13.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.3T + 6.89e4T^{2} \) |
| 47 | \( 1 - 409.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 706.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 334.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 480.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 250.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 351.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 609.T + 9.12e5T^{2} \) |
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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
−8.238947634279012732059019679816, −7.46185651815563519601966168493, −6.90980520199659170477671994255, −6.02863745184823709003527684712, −4.78112406687328918023496456493, −4.35352290485521904412448907285, −3.35115134683885930938649563608, −2.80018995284851445899769535569, −2.24606628322871973689621728452, 0,
2.24606628322871973689621728452, 2.80018995284851445899769535569, 3.35115134683885930938649563608, 4.35352290485521904412448907285, 4.78112406687328918023496456493, 6.02863745184823709003527684712, 6.90980520199659170477671994255, 7.46185651815563519601966168493, 8.238947634279012732059019679816