| L(s) = 1 | + 4.85·2-s + 5.73·3-s + 15.6·4-s − 14.4·5-s + 27.8·6-s + 13.1·7-s + 36.9·8-s + 5.83·9-s − 70.2·10-s − 56.6·11-s + 89.4·12-s − 68.5·13-s + 64.0·14-s − 82.8·15-s + 54.6·16-s + 23.3·17-s + 28.3·18-s + 119.·19-s − 225.·20-s + 75.5·21-s − 275.·22-s − 49.8·23-s + 211.·24-s + 84.0·25-s − 333.·26-s − 121.·27-s + 205.·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.10·3-s + 1.95·4-s − 1.29·5-s + 1.89·6-s + 0.711·7-s + 1.63·8-s + 0.216·9-s − 2.22·10-s − 1.55·11-s + 2.15·12-s − 1.46·13-s + 1.22·14-s − 1.42·15-s + 0.854·16-s + 0.333·17-s + 0.371·18-s + 1.44·19-s − 2.52·20-s + 0.785·21-s − 2.66·22-s − 0.451·23-s + 1.80·24-s + 0.672·25-s − 2.51·26-s − 0.864·27-s + 1.38·28-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 | \( 1 - 4.85T + 8T^{2} \) |
| 3 | \( 1 - 5.73T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 - 13.1T + 343T^{2} \) |
| 11 | \( 1 + 56.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 290.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 712.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 70.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 328.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 307.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 203.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 255.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 665.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 834.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 817.T + 9.12e5T^{2} \) |
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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
−7.939824341677443617121289383881, −7.69367398286137394747124269935, −7.19946166840507711154189242043, −5.56609088703234684620144734653, −5.13237917634236589654555882680, −4.26414595058166283492306174403, −3.44728438858278426201420470457, −2.81747140580826176073086396127, −2.03878487370906251138983799389, 0,
2.03878487370906251138983799389, 2.81747140580826176073086396127, 3.44728438858278426201420470457, 4.26414595058166283492306174403, 5.13237917634236589654555882680, 5.56609088703234684620144734653, 7.19946166840507711154189242043, 7.69367398286137394747124269935, 7.939824341677443617121289383881
