L(s) = 1 | + 0.414·2-s − 7.82·4-s + 0.100·5-s − 6.55·8-s + 0.0416·10-s + 43.9·11-s − 16.6·13-s + 59.9·16-s + 121.·17-s − 127.·19-s − 0.786·20-s + 18.2·22-s − 53.5·23-s − 124.·25-s − 6.89·26-s − 235.·29-s − 18.7·31-s + 77.2·32-s + 50.3·34-s − 191.·37-s − 52.6·38-s − 0.658·40-s + 319.·41-s − 218.·43-s − 343.·44-s − 22.2·46-s − 401.·47-s + ⋯ |
L(s) = 1 | + 0.146·2-s − 0.978·4-s + 0.00898·5-s − 0.289·8-s + 0.00131·10-s + 1.20·11-s − 0.355·13-s + 0.936·16-s + 1.73·17-s − 1.53·19-s − 0.00879·20-s + 0.176·22-s − 0.485·23-s − 0.999·25-s − 0.0520·26-s − 1.50·29-s − 0.108·31-s + 0.426·32-s + 0.254·34-s − 0.852·37-s − 0.224·38-s − 0.00260·40-s + 1.21·41-s − 0.775·43-s − 1.17·44-s − 0.0711·46-s − 1.24·47-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 | \( 1 - 0.414T + 8T^{2} \) |
| 5 | \( 1 - 0.100T + 125T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 11.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 12.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 805.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 394.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−9.969466995847684466964555341501, −9.476274308523786169016516636709, −8.474033469277892391220969697937, −7.63773511248203626881775284757, −6.31024770635862765524137790211, −5.42997434334579111917571478556, −4.23526549429974276304580344014, −3.48169475595731462859201874757, −1.61028121729515127856046761135, 0,
1.61028121729515127856046761135, 3.48169475595731462859201874757, 4.23526549429974276304580344014, 5.42997434334579111917571478556, 6.31024770635862765524137790211, 7.63773511248203626881775284757, 8.474033469277892391220969697937, 9.476274308523786169016516636709, 9.969466995847684466964555341501