L(s) = 1 | + 5-s + 2·7-s + 13-s + 4·17-s + 4·19-s + 23-s + 25-s + 3·29-s + 31-s + 2·35-s − 8·37-s + 5·41-s + 6·43-s + 9·47-s − 3·49-s − 2·53-s + 65-s − 4·67-s + 3·71-s + 7·73-s − 4·79-s + 8·83-s + 4·85-s + 14·89-s + 2·91-s + 4·95-s − 14·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.277·13-s + 0.970·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.557·29-s + 0.179·31-s + 0.338·35-s − 1.31·37-s + 0.780·41-s + 0.914·43-s + 1.31·47-s − 3/7·49-s − 0.274·53-s + 0.124·65-s − 0.488·67-s + 0.356·71-s + 0.819·73-s − 0.450·79-s + 0.878·83-s + 0.433·85-s + 1.48·89-s + 0.209·91-s + 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.242450267\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.242450267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.96157215212702, −15.38376284000489, −14.70723088470239, −14.15988768256748, −13.88150713921307, −13.26580346132684, −12.39249800441510, −12.15759204525647, −11.42739304886733, −10.81178046615915, −10.36982914993156, −9.652106598472096, −9.146260906117347, −8.487223093278698, −7.831821652892069, −7.373783361569030, −6.610875623920605, −5.858331685041043, −5.334540082211549, −4.776202683622259, −3.922829966450851, −3.184719620683462, −2.411493650569177, −1.499060914972128, −0.8414880443943651,
0.8414880443943651, 1.499060914972128, 2.411493650569177, 3.184719620683462, 3.922829966450851, 4.776202683622259, 5.334540082211549, 5.858331685041043, 6.610875623920605, 7.373783361569030, 7.831821652892069, 8.487223093278698, 9.146260906117347, 9.652106598472096, 10.36982914993156, 10.81178046615915, 11.42739304886733, 12.15759204525647, 12.39249800441510, 13.26580346132684, 13.88150713921307, 14.15988768256748, 14.70723088470239, 15.38376284000489, 15.96157215212702