| L(s) = 1 | − 2·3-s − 6·5-s + 20·7-s − 23·9-s − 14·11-s − 54·13-s + 12·15-s − 66·17-s − 162·19-s − 40·21-s + 172·23-s − 89·25-s + 100·27-s + 2·29-s − 128·31-s + 28·33-s − 120·35-s − 158·37-s + 108·39-s + 202·41-s + 298·43-s + 138·45-s − 408·47-s + 57·49-s + 132·51-s + 690·53-s + 84·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.536·5-s + 1.07·7-s − 0.851·9-s − 0.383·11-s − 1.15·13-s + 0.206·15-s − 0.941·17-s − 1.95·19-s − 0.415·21-s + 1.55·23-s − 0.711·25-s + 0.712·27-s + 0.0128·29-s − 0.741·31-s + 0.147·33-s − 0.579·35-s − 0.702·37-s + 0.443·39-s + 0.769·41-s + 1.05·43-s + 0.457·45-s − 1.26·47-s + 0.166·49-s + 0.362·51-s + 1.78·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 162 T + p^{3} T^{2} \) |
| 23 | \( 1 - 172 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 - 298 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 690 T + p^{3} T^{2} \) |
| 59 | \( 1 - 322 T + p^{3} T^{2} \) |
| 61 | \( 1 - 298 T + p^{3} T^{2} \) |
| 67 | \( 1 + 202 T + p^{3} T^{2} \) |
| 71 | \( 1 + 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 744 T + p^{3} T^{2} \) |
| 83 | \( 1 - 678 T + p^{3} T^{2} \) |
| 89 | \( 1 + 82 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1122 T + p^{3} T^{2} \) |
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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
−12.20580084433307903650585973692, −11.23927000647250213360938006532, −10.67585939278352530972271723791, −8.972751277221209892132498160359, −8.087932411560379469880011895417, −6.90590838385910866493080894159, −5.38390579785305732936730283713, −4.37374902352165279091785597577, −2.36703794839383207403309880984, 0,
2.36703794839383207403309880984, 4.37374902352165279091785597577, 5.38390579785305732936730283713, 6.90590838385910866493080894159, 8.087932411560379469880011895417, 8.972751277221209892132498160359, 10.67585939278352530972271723791, 11.23927000647250213360938006532, 12.20580084433307903650585973692
