L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 2·11-s − 13-s − 3·17-s − 19-s + 3·21-s − 23-s − 5·27-s + 5·29-s + 8·31-s + 2·33-s − 2·37-s − 39-s − 8·41-s − 4·43-s + 8·47-s + 2·49-s − 3·51-s − 53-s − 57-s + 15·59-s − 2·61-s − 6·63-s − 3·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s − 0.727·17-s − 0.229·19-s + 0.654·21-s − 0.208·23-s − 0.962·27-s + 0.928·29-s + 1.43·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 0.609·43-s + 1.16·47-s + 2/7·49-s − 0.420·51-s − 0.137·53-s − 0.132·57-s + 1.95·59-s − 0.256·61-s − 0.755·63-s − 0.366·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.980082933\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.980082933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.90339321446824, −14.66777581108831, −14.10729093965212, −13.56241777672785, −13.34071516051521, −12.16259956250201, −12.02418162963794, −11.41920689694339, −10.93044883597597, −10.24234780410984, −9.746080589168015, −8.913076868531012, −8.554259102762212, −8.224212309261141, −7.555633231031295, −6.831564619222120, −6.322312261738303, −5.558233479117131, −4.874537810034686, −4.409919735226398, −3.678973835590199, −2.900297734242442, −2.248822231069611, −1.628961382686736, −0.6343155773335693,
0.6343155773335693, 1.628961382686736, 2.248822231069611, 2.900297734242442, 3.678973835590199, 4.409919735226398, 4.874537810034686, 5.558233479117131, 6.322312261738303, 6.831564619222120, 7.555633231031295, 8.224212309261141, 8.554259102762212, 8.913076868531012, 9.746080589168015, 10.24234780410984, 10.93044883597597, 11.41920689694339, 12.02418162963794, 12.16259956250201, 13.34071516051521, 13.56241777672785, 14.10729093965212, 14.66777581108831, 14.90339321446824