L(s) = 1 | + 2·2-s + 4·4-s + 4·7-s + 8·8-s − 42·11-s − 20·13-s + 8·14-s + 16·16-s + 93·17-s + 59·19-s − 84·22-s + 9·23-s − 40·26-s + 16·28-s − 120·29-s + 47·31-s + 32·32-s + 186·34-s + 262·37-s + 118·38-s − 126·41-s + 178·43-s − 168·44-s + 18·46-s + 144·47-s − 327·49-s − 80·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.215·7-s + 0.353·8-s − 1.15·11-s − 0.426·13-s + 0.152·14-s + 1/4·16-s + 1.32·17-s + 0.712·19-s − 0.814·22-s + 0.0815·23-s − 0.301·26-s + 0.107·28-s − 0.768·29-s + 0.272·31-s + 0.176·32-s + 0.938·34-s + 1.16·37-s + 0.503·38-s − 0.479·41-s + 0.631·43-s − 0.575·44-s + 0.0576·46-s + 0.446·47-s − 0.953·49-s − 0.213·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.435746157\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.435746157\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 - 59 T + p^{3} T^{2} \) |
| 23 | \( 1 - 9 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 47 T + p^{3} T^{2} \) |
| 37 | \( 1 - 262 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 741 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 221 T + p^{3} T^{2} \) |
| 67 | \( 1 - 538 T + p^{3} T^{2} \) |
| 71 | \( 1 + 690 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1126 T + p^{3} T^{2} \) |
| 79 | \( 1 - 665 T + p^{3} T^{2} \) |
| 83 | \( 1 - 75 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1544 T + p^{3} T^{2} \) |
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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
−9.403047305805344839122838368938, −8.087034733027938236516159384912, −7.66702847130679060421373638112, −6.76338228714600270939140875910, −5.51388039446001192140839301745, −5.28224078887667504905667824373, −4.10137778529032523346201001899, −3.10971591157685679855766932035, −2.23505131907932958020358425295, −0.836470127643280600879351906015,
0.836470127643280600879351906015, 2.23505131907932958020358425295, 3.10971591157685679855766932035, 4.10137778529032523346201001899, 5.28224078887667504905667824373, 5.51388039446001192140839301745, 6.76338228714600270939140875910, 7.66702847130679060421373638112, 8.087034733027938236516159384912, 9.403047305805344839122838368938