| L(s) = 1 | + 7·3-s − 34·7-s + 22·9-s − 27·11-s + 28·13-s − 21·17-s − 35·19-s − 238·21-s − 78·23-s − 35·27-s − 120·29-s − 182·31-s − 189·33-s − 146·37-s + 196·39-s + 357·41-s − 148·43-s − 84·47-s + 813·49-s − 147·51-s − 702·53-s − 245·57-s + 840·59-s − 238·61-s − 748·63-s + 461·67-s − 546·69-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 1.83·7-s + 0.814·9-s − 0.740·11-s + 0.597·13-s − 0.299·17-s − 0.422·19-s − 2.47·21-s − 0.707·23-s − 0.249·27-s − 0.768·29-s − 1.05·31-s − 0.996·33-s − 0.648·37-s + 0.804·39-s + 1.35·41-s − 0.524·43-s − 0.260·47-s + 2.37·49-s − 0.403·51-s − 1.81·53-s − 0.569·57-s + 1.85·59-s − 0.499·61-s − 1.49·63-s + 0.840·67-s − 0.952·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 35 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 + 182 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 357 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 840 T + p^{3} T^{2} \) |
| 61 | \( 1 + 238 T + p^{3} T^{2} \) |
| 67 | \( 1 - 461 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 - 133 T + p^{3} T^{2} \) |
| 79 | \( 1 + 650 T + p^{3} T^{2} \) |
| 83 | \( 1 + 903 T + p^{3} T^{2} \) |
| 89 | \( 1 - 735 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1106 T + p^{3} T^{2} \) |
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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
−10.08378221014869929720460564273, −9.416057280281955953596376254134, −8.676545811088546017954879730489, −7.72361713912021358065088015040, −6.71187077932992454421572873710, −5.70773194360831706309944798381, −3.92013347004817529016784796320, −3.20823583862386584754947896184, −2.19092474485564712267481820880, 0,
2.19092474485564712267481820880, 3.20823583862386584754947896184, 3.92013347004817529016784796320, 5.70773194360831706309944798381, 6.71187077932992454421572873710, 7.72361713912021358065088015040, 8.676545811088546017954879730489, 9.416057280281955953596376254134, 10.08378221014869929720460564273
