| L(s) = 1 | + 1.84·2-s − 0.727·3-s − 4.59·4-s − 12.1·5-s − 1.34·6-s + 17.8·7-s − 23.2·8-s − 26.4·9-s − 22.4·10-s + 16.9·11-s + 3.34·12-s − 51.5·13-s + 32.9·14-s + 8.84·15-s − 6.07·16-s + 120.·17-s − 48.8·18-s + 5.77·19-s + 55.9·20-s − 12.9·21-s + 31.3·22-s + 210.·23-s + 16.9·24-s + 22.8·25-s − 95.0·26-s + 38.9·27-s − 82.0·28-s + ⋯ |
| L(s) = 1 | + 0.652·2-s − 0.140·3-s − 0.574·4-s − 1.08·5-s − 0.0913·6-s + 0.963·7-s − 1.02·8-s − 0.980·9-s − 0.709·10-s + 0.465·11-s + 0.0804·12-s − 1.09·13-s + 0.628·14-s + 0.152·15-s − 0.0949·16-s + 1.72·17-s − 0.639·18-s + 0.0697·19-s + 0.624·20-s − 0.134·21-s + 0.303·22-s + 1.90·23-s + 0.143·24-s + 0.182·25-s − 0.716·26-s + 0.277·27-s − 0.553·28-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 | \( 1 - 1.84T + 8T^{2} \) |
| 3 | \( 1 + 0.727T + 27T^{2} \) |
| 5 | \( 1 + 12.1T + 125T^{2} \) |
| 7 | \( 1 - 17.8T + 343T^{2} \) |
| 11 | \( 1 - 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.77T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 12.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 233.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 92.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 190.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 62.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 564.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 339.T + 9.12e5T^{2} \) |
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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
−8.390284971747552794385373180885, −7.78197124796209801986282752207, −7.00784729362665704977533749184, −5.66936406723294796589216153993, −5.17779350344858582512303452732, −4.45365660449633502537736762558, −3.52757052669789767003116393891, −2.81272763601038546506894616079, −1.08723167408126534966198901374, 0,
1.08723167408126534966198901374, 2.81272763601038546506894616079, 3.52757052669789767003116393891, 4.45365660449633502537736762558, 5.17779350344858582512303452732, 5.66936406723294796589216153993, 7.00784729362665704977533749184, 7.78197124796209801986282752207, 8.390284971747552794385373180885
