| L(s) = 1 | − 9.45·3-s + 11.5·5-s − 15.9·7-s + 62.4·9-s − 19.4·11-s + 69.4·13-s − 109.·15-s − 112.·17-s − 19·19-s + 150.·21-s − 57.3·23-s + 9.43·25-s − 334.·27-s + 59.8·29-s + 222.·31-s + 183.·33-s − 184.·35-s + 258.·37-s − 657.·39-s − 305.·41-s + 322.·43-s + 723.·45-s + 213.·47-s − 90.0·49-s + 1.06e3·51-s − 348.·53-s − 225.·55-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 1.03·5-s − 0.858·7-s + 2.31·9-s − 0.532·11-s + 1.48·13-s − 1.88·15-s − 1.60·17-s − 0.229·19-s + 1.56·21-s − 0.520·23-s + 0.0754·25-s − 2.38·27-s + 0.383·29-s + 1.29·31-s + 0.968·33-s − 0.890·35-s + 1.14·37-s − 2.69·39-s − 1.16·41-s + 1.14·43-s + 2.39·45-s + 0.663·47-s − 0.262·49-s + 2.91·51-s − 0.903·53-s − 0.551·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 9.45T + 27T^{2} \) |
| 5 | \( 1 - 11.5T + 125T^{2} \) |
| 7 | \( 1 + 15.9T + 343T^{2} \) |
| 11 | \( 1 + 19.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 57.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 59.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 348.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 56.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 87.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 725.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 889.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 227.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 927.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 603.T + 9.12e5T^{2} \) |
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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
−9.263031846785968774109458368628, −8.101008016844392912638518998882, −6.62968783463327343211495983631, −6.40120075376274004808687045007, −5.81058997473476514614615608881, −4.87651204911010981363461444724, −3.94696287435849910027315431964, −2.35488832904644812261851536123, −1.11033394597068527769673712878, 0,
1.11033394597068527769673712878, 2.35488832904644812261851536123, 3.94696287435849910027315431964, 4.87651204911010981363461444724, 5.81058997473476514614615608881, 6.40120075376274004808687045007, 6.62968783463327343211495983631, 8.101008016844392912638518998882, 9.263031846785968774109458368628
