| L(s) = 1 | − 5.56·2-s + 23.0·4-s − 5.10·5-s − 6.75·7-s − 83.5·8-s + 28.4·10-s + 53.8·11-s − 45.8·13-s + 37.6·14-s + 281.·16-s + 17·17-s + 93.8·19-s − 117.·20-s − 299.·22-s − 40.2·23-s − 98.9·25-s + 255.·26-s − 155.·28-s − 224.·29-s − 235.·31-s − 896.·32-s − 94.6·34-s + 34.4·35-s − 203.·37-s − 522.·38-s + 426.·40-s + 245.·41-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.87·4-s − 0.456·5-s − 0.364·7-s − 3.69·8-s + 0.898·10-s + 1.47·11-s − 0.978·13-s + 0.718·14-s + 4.39·16-s + 0.242·17-s + 1.13·19-s − 1.31·20-s − 2.90·22-s − 0.365·23-s − 0.791·25-s + 1.92·26-s − 1.04·28-s − 1.43·29-s − 1.36·31-s − 4.95·32-s − 0.477·34-s + 0.166·35-s − 0.905·37-s − 2.23·38-s + 1.68·40-s + 0.936·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 5.56T + 8T^{2} \) |
| 5 | \( 1 + 5.10T + 125T^{2} \) |
| 7 | \( 1 + 6.75T + 343T^{2} \) |
| 11 | \( 1 - 53.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 93.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 235.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 56.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 739.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 895.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 182.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 613.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 289.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 516.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 643.T + 9.12e5T^{2} \) |
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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
−11.73764348639962444439228778057, −10.81343556318154555043171253030, −9.446034263769245457050698780613, −9.325757825910972159457224351435, −7.78373428197856523003261724800, −7.18642299157098383733562004188, −5.97086716401058306738503534112, −3.41460676380534182804424992030, −1.66107391815441758875848638956, 0,
1.66107391815441758875848638956, 3.41460676380534182804424992030, 5.97086716401058306738503534112, 7.18642299157098383733562004188, 7.78373428197856523003261724800, 9.325757825910972159457224351435, 9.446034263769245457050698780613, 10.81343556318154555043171253030, 11.73764348639962444439228778057
