| L(s) = 1 | + 3-s − 2·5-s − 7-s − 2·9-s + 2·11-s + 13-s − 2·15-s + 6·19-s − 21-s − 25-s − 5·27-s − 29-s − 31-s + 2·33-s + 2·35-s − 6·37-s + 39-s + 3·41-s + 4·45-s + 3·47-s + 49-s + 6·53-s − 4·55-s + 6·57-s + 8·59-s − 10·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 1.37·19-s − 0.218·21-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.160·39-s + 0.468·41-s + 0.596·45-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 0.794·57-s + 1.04·59-s − 1.28·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.26143988813975, −12.59177298965122, −12.07574023669827, −11.73788403003714, −11.49275353707114, −10.81393464992706, −10.41887646951838, −9.728710198698621, −9.298252490906837, −8.916017917028031, −8.524960001226065, −7.828921475467155, −7.627786868544083, −7.107515638577163, −6.496632601458394, −5.982838381192041, −5.411426138338792, −4.984185717378922, −4.103336164164820, −3.777082775788056, −3.369002611284707, −2.839867963274763, −2.222722484593977, −1.453600443087633, −0.7435797800654889, 0,
0.7435797800654889, 1.453600443087633, 2.222722484593977, 2.839867963274763, 3.369002611284707, 3.777082775788056, 4.103336164164820, 4.984185717378922, 5.411426138338792, 5.982838381192041, 6.496632601458394, 7.107515638577163, 7.627786868544083, 7.828921475467155, 8.524960001226065, 8.916017917028031, 9.298252490906837, 9.728710198698621, 10.41887646951838, 10.81393464992706, 11.49275353707114, 11.73788403003714, 12.07574023669827, 12.59177298965122, 13.26143988813975
