L(s) = 1 | + 5-s + (1.80 − 1.93i)7-s − 3.87i·11-s − 1.60i·13-s + 8.11·17-s − 2.63i·19-s − 5.47i·23-s + 25-s + 5.47i·29-s + 3.73i·31-s + (1.80 − 1.93i)35-s + 4.51·37-s − 1.60·41-s + 10.1·43-s − 11.1·47-s + ⋯ |
L(s) = 1 | + 0.447·5-s + (0.681 − 0.732i)7-s − 1.16i·11-s − 0.445i·13-s + 1.96·17-s − 0.605i·19-s − 1.14i·23-s + 0.200·25-s + 1.01i·29-s + 0.670i·31-s + (0.304 − 0.327i)35-s + 0.741·37-s − 0.250·41-s + 1.54·43-s − 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.471572908\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.471572908\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + (-1.80 + 1.93i)T \) |
good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 + 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 8.11T + 17T^{2} \) |
| 19 | \( 1 + 2.63iT - 19T^{2} \) |
| 23 | \( 1 + 5.47iT - 23T^{2} \) |
| 29 | \( 1 - 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.90T + 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−8.063172099870126577835743007861, −7.47638474257819870296035924694, −6.62879029520023733290492443918, −5.82408801605981134839077995150, −5.22525476409003704839103862040, −4.45153864073803273780109078957, −3.39629332178253487041523263485, −2.83720419662562984824446166176, −1.44661408236960861991290851304, −0.71797308879832407142659111759,
1.32363242293354653409527636398, 1.99221143326514375872025132929, 2.93909982400692383835325587545, 3.99290239663349043591700654632, 4.78437442591239767537848390157, 5.60968033736888232866102943803, 5.97452930391338128336946074745, 7.06331978840094000934457419146, 7.83926291362757708739675067280, 8.126902206483069108146363316441