L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 2·11-s + 2·13-s + 16-s − 4·17-s + 6·19-s + 20-s + 2·22-s − 6·23-s + 25-s + 2·26-s + 10·29-s + 8·31-s + 32-s − 4·34-s + 2·37-s + 6·38-s + 40-s + 2·41-s − 43-s + 2·44-s − 6·46-s − 2·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s + 1.37·19-s + 0.223·20-s + 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 1.85·29-s + 1.43·31-s + 0.176·32-s − 0.685·34-s + 0.328·37-s + 0.973·38-s + 0.158·40-s + 0.312·41-s − 0.152·43-s + 0.301·44-s − 0.884·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.35501424115304, −13.08184848425551, −12.33495135637256, −11.94919300104427, −11.52864723343728, −11.27796741517843, −10.34194244600348, −10.20459214422116, −9.704820337551586, −9.032999901917924, −8.584791672099778, −8.121138613290175, −7.499736424571289, −6.947273012680609, −6.385606906499327, −6.132778341222181, −5.632623248743064, −4.911631929766750, −4.428443864157054, −4.112739821736905, −3.287094133688472, −2.825245330201031, −2.345478131284805, −1.332328009906875, −1.236156409410615, 0,
1.236156409410615, 1.332328009906875, 2.345478131284805, 2.825245330201031, 3.287094133688472, 4.112739821736905, 4.428443864157054, 4.911631929766750, 5.632623248743064, 6.132778341222181, 6.385606906499327, 6.947273012680609, 7.499736424571289, 8.121138613290175, 8.584791672099778, 9.032999901917924, 9.704820337551586, 10.20459214422116, 10.34194244600348, 11.27796741517843, 11.52864723343728, 11.94919300104427, 12.33495135637256, 13.08184848425551, 13.35501424115304