| L(s) = 1 | − 8·4-s − 18·5-s + 64·16-s + 144·20-s + 108·23-s + 199·25-s + 340·31-s − 434·37-s + 36·47-s − 343·49-s + 738·53-s + 720·59-s − 512·64-s − 416·67-s − 612·71-s − 1.15e3·80-s − 1.67e3·89-s − 864·92-s − 34·97-s − 1.59e3·100-s + 1.17e3·103-s − 2.14e3·113-s − 1.94e3·115-s + ⋯ |
| L(s) = 1 | − 4-s − 1.60·5-s + 16-s + 1.60·20-s + 0.979·23-s + 1.59·25-s + 1.96·31-s − 1.92·37-s + 0.111·47-s − 49-s + 1.91·53-s + 1.58·59-s − 64-s − 0.758·67-s − 1.02·71-s − 1.60·80-s − 1.99·89-s − 0.979·92-s − 0.0355·97-s − 1.59·100-s + 1.12·103-s − 1.78·113-s − 1.57·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 - 108 T + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 340 T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 - 36 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 - 720 T + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 + 416 T + p^{3} T^{2} \) |
| 71 | \( 1 + 612 T + p^{3} T^{2} \) |
| 73 | \( 1 + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + 1674 T + p^{3} T^{2} \) |
| 97 | \( 1 + 34 T + p^{3} T^{2} \) |
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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
−8.736785677830695573996941291180, −8.446377138974293221095131063340, −7.53715153686217732863227816235, −6.75574963478171206685547455048, −5.37322498552637436041865688150, −4.56837298844257880324597562311, −3.85429552969987359634357864643, −2.98709261059238423344664646162, −1.00767604415401823796887979437, 0,
1.00767604415401823796887979437, 2.98709261059238423344664646162, 3.85429552969987359634357864643, 4.56837298844257880324597562311, 5.37322498552637436041865688150, 6.75574963478171206685547455048, 7.53715153686217732863227816235, 8.446377138974293221095131063340, 8.736785677830695573996941291180
