| 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
−12.49250927433566144823884122458, −11.19309127772426982466805386898, −10.38419455022517089481749640489, −9.204753736933183415426676755629, −8.194807636114177184211926510410, −6.28415107340085725171821870300, −6.03729805240788441996694338466, −4.11908511555623072926490989423, −2.43405629277512544012426618019, 0,
2.43405629277512544012426618019, 4.11908511555623072926490989423, 6.03729805240788441996694338466, 6.28415107340085725171821870300, 8.194807636114177184211926510410, 9.204753736933183415426676755629, 10.38419455022517089481749640489, 11.19309127772426982466805386898, 12.49250927433566144823884122458
