| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 2·11-s + 13-s − 15-s + 17-s + 6·19-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 4·31-s + 2·33-s − 4·35-s + 8·37-s − 39-s + 6·41-s + 8·43-s + 45-s + 9·49-s − 51-s + 6·53-s − 2·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.676·35-s + 1.31·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 9/7·49-s − 0.140·51-s + 0.824·53-s − 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.20800839245593, −12.78904502327966, −12.30573137220344, −11.91649918663091, −11.46999320012174, −10.78902344000035, −10.29076286989869, −10.05297289129807, −9.597554949914093, −9.138352067230433, −8.656189276210736, −7.819862426730263, −7.510203179649901, −7.028583165136749, −6.198723742516188, −6.090203351038182, −5.767519566751232, −4.993827588745534, −4.489889746811468, −3.898014293835647, −3.188994834166889, −2.797731795013649, −2.302780124856013, −1.230422430313376, −0.8130640934663036, 0,
0.8130640934663036, 1.230422430313376, 2.302780124856013, 2.797731795013649, 3.188994834166889, 3.898014293835647, 4.489889746811468, 4.993827588745534, 5.767519566751232, 6.090203351038182, 6.198723742516188, 7.028583165136749, 7.510203179649901, 7.819862426730263, 8.656189276210736, 9.138352067230433, 9.597554949914093, 10.05297289129807, 10.29076286989869, 10.78902344000035, 11.46999320012174, 11.91649918663091, 12.30573137220344, 12.78904502327966, 13.20800839245593
