| L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·13-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s − 4·27-s + 6·29-s + 4·31-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s + 12·47-s + 49-s − 12·51-s + 6·53-s + 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s − 2·73-s − 10·75-s + 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.234·73-s − 1.15·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.584382463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584382463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.88587571481399, −15.50665846206611, −15.25987665815362, −14.34796181156849, −13.91545321327049, −13.54925317898306, −13.13321746227164, −12.19370025397593, −11.67395523591299, −11.02709474631607, −10.51810525643189, −9.710198779582612, −9.139513650481559, −8.527724601400848, −8.305301817736640, −7.527253091226246, −6.879758104044883, −6.101217494994411, −5.514875107512378, −4.427895610575064, −4.057318011322368, −3.206712236945471, −2.498224450097883, −1.862509608743078, −0.7959139318593843,
0.7959139318593843, 1.862509608743078, 2.498224450097883, 3.206712236945471, 4.057318011322368, 4.427895610575064, 5.514875107512378, 6.101217494994411, 6.879758104044883, 7.527253091226246, 8.305301817736640, 8.527724601400848, 9.139513650481559, 9.710198779582612, 10.51810525643189, 11.02709474631607, 11.67395523591299, 12.19370025397593, 13.13321746227164, 13.54925317898306, 13.91545321327049, 14.34796181156849, 15.25987665815362, 15.50665846206611, 15.88587571481399
