L(s) = 1 | − 4.53·2-s + 12.5·4-s + 13.4·5-s − 20.5·8-s − 61.0·10-s − 0.813·11-s + 34.9·13-s − 7.21·16-s − 117.·17-s − 93.2·19-s + 168.·20-s + 3.68·22-s − 120.·23-s + 56.6·25-s − 158.·26-s − 8.56·29-s − 82.1·31-s + 196.·32-s + 533.·34-s + 28.8·37-s + 422.·38-s − 276.·40-s + 70.5·41-s + 417.·43-s − 10.1·44-s + 544.·46-s − 338.·47-s + ⋯ |
L(s) = 1 | − 1.60·2-s + 1.56·4-s + 1.20·5-s − 0.907·8-s − 1.93·10-s − 0.0222·11-s + 0.745·13-s − 0.112·16-s − 1.67·17-s − 1.12·19-s + 1.88·20-s + 0.0357·22-s − 1.09·23-s + 0.453·25-s − 1.19·26-s − 0.0548·29-s − 0.475·31-s + 1.08·32-s + 2.69·34-s + 0.128·37-s + 1.80·38-s − 1.09·40-s + 0.268·41-s + 1.47·43-s − 0.0349·44-s + 1.74·46-s − 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 4.53T + 8T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 11 | \( 1 + 0.813T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.56T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 94.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 792.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 550.T + 9.12e5T^{2} \) |
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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
−10.19625774997394633204641230754, −9.172912802797742768093577994576, −8.789823239679711789293766733907, −7.74081695135161791105500974679, −6.55856877070520430346582085119, −5.99551690714929839941436784478, −4.34714868075364767883805524282, −2.38415776272345090808390566435, −1.61262597370644983258921183442, 0,
1.61262597370644983258921183442, 2.38415776272345090808390566435, 4.34714868075364767883805524282, 5.99551690714929839941436784478, 6.55856877070520430346582085119, 7.74081695135161791105500974679, 8.789823239679711789293766733907, 9.172912802797742768093577994576, 10.19625774997394633204641230754