| L(s) = 1 | − 4.38·2-s − 3·3-s + 11.2·4-s + 5·5-s + 13.1·6-s − 11.7·7-s − 14.2·8-s + 9·9-s − 21.9·10-s + 11·11-s − 33.7·12-s − 72.8·13-s + 51.4·14-s − 15·15-s − 27.3·16-s − 9.89·17-s − 39.4·18-s + 0.0238·19-s + 56.2·20-s + 35.1·21-s − 48.2·22-s + 73.0·23-s + 42.8·24-s + 25·25-s + 319.·26-s − 27·27-s − 132.·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.577·3-s + 1.40·4-s + 0.447·5-s + 0.895·6-s − 0.633·7-s − 0.631·8-s + 0.333·9-s − 0.693·10-s + 0.301·11-s − 0.812·12-s − 1.55·13-s + 0.982·14-s − 0.258·15-s − 0.426·16-s − 0.141·17-s − 0.517·18-s + 0.000288·19-s + 0.629·20-s + 0.365·21-s − 0.467·22-s + 0.662·23-s + 0.364·24-s + 0.200·25-s + 2.41·26-s − 0.192·27-s − 0.891·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5458311682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5458311682\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 4.38T + 8T^{2} \) |
| 7 | \( 1 + 11.7T + 343T^{2} \) |
| 13 | \( 1 + 72.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.89T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.0238T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 88.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 347.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 491.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 159.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 567.T + 9.12e5T^{2} \) |
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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
−12.07533936237075408144944627916, −11.07904444892634687709549286026, −9.909103159308129748900987866630, −9.684556257828984750693164202204, −8.396144790289119131559845953362, −7.17114604733085893981143726309, −6.39080057052814645048120108073, −4.79802128578515615762823476461, −2.49641627590220521141409036631, −0.75331159580547719371251200662,
0.75331159580547719371251200662, 2.49641627590220521141409036631, 4.79802128578515615762823476461, 6.39080057052814645048120108073, 7.17114604733085893981143726309, 8.396144790289119131559845953362, 9.684556257828984750693164202204, 9.909103159308129748900987866630, 11.07904444892634687709549286026, 12.07533936237075408144944627916
