| L(s) = 1 | − 12.7·3-s − 83.6·5-s + 121.·7-s − 80.9·9-s + 227.·11-s − 501.·13-s + 1.06e3·15-s − 1.30e3·17-s + 23.0·19-s − 1.54e3·21-s − 1.19e3·23-s + 3.87e3·25-s + 4.12e3·27-s + 4.35e3·29-s + 4.11e3·31-s − 2.89e3·33-s − 1.01e4·35-s + 4.01e3·37-s + 6.39e3·39-s − 893.·41-s − 1.31e4·43-s + 6.76e3·45-s − 1.87e4·47-s − 2.02e3·49-s + 1.65e4·51-s + 1.53e3·53-s − 1.90e4·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.49·5-s + 0.937·7-s − 0.332·9-s + 0.567·11-s − 0.823·13-s + 1.22·15-s − 1.09·17-s + 0.0146·19-s − 0.766·21-s − 0.470·23-s + 1.23·25-s + 1.08·27-s + 0.961·29-s + 0.769·31-s − 0.463·33-s − 1.40·35-s + 0.481·37-s + 0.672·39-s − 0.0830·41-s − 1.08·43-s + 0.498·45-s − 1.23·47-s − 0.120·49-s + 0.892·51-s + 0.0748·53-s − 0.848·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 12.7T + 243T^{2} \) |
| 5 | \( 1 + 83.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 121.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 227.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 501.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 23.0T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 893.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.31e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.51e4T + 8.58e9T^{2} \) |
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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
−8.360154574204342112454637478596, −8.207362489349234163868085926458, −7.03095953341056625600428967880, −6.39358336789857909771831138119, −5.03242753511244873985554255383, −4.61584918245837985564599943166, −3.62538647628427462132853202559, −2.32426824376478280632400973541, −0.868221724695908888340063522196, 0,
0.868221724695908888340063522196, 2.32426824376478280632400973541, 3.62538647628427462132853202559, 4.61584918245837985564599943166, 5.03242753511244873985554255383, 6.39358336789857909771831138119, 7.03095953341056625600428967880, 8.207362489349234163868085926458, 8.360154574204342112454637478596
