L(s) = 1 | + 6.55·2-s − 9·3-s + 11.0·4-s + 25·5-s − 59.0·6-s + 146.·7-s − 137.·8-s + 81·9-s + 163.·10-s − 121·11-s − 99.1·12-s + 170.·13-s + 960.·14-s − 225·15-s − 1.25e3·16-s + 1.56e3·17-s + 531.·18-s + 569.·19-s + 275.·20-s − 1.31e3·21-s − 793.·22-s + 3.15e3·23-s + 1.23e3·24-s + 625·25-s + 1.11e3·26-s − 729·27-s + 1.61e3·28-s + ⋯ |
L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.344·4-s + 0.447·5-s − 0.669·6-s + 1.12·7-s − 0.760·8-s + 0.333·9-s + 0.518·10-s − 0.301·11-s − 0.198·12-s + 0.279·13-s + 1.31·14-s − 0.258·15-s − 1.22·16-s + 1.31·17-s + 0.386·18-s + 0.362·19-s + 0.153·20-s − 0.652·21-s − 0.349·22-s + 1.24·23-s + 0.438·24-s + 0.200·25-s + 0.323·26-s − 0.192·27-s + 0.389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.401646108\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.401646108\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 6.55T + 32T^{2} \) |
| 7 | \( 1 - 146.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 170.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 569.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.85e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.64e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.58e4T + 8.58e9T^{2} \) |
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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
−12.04557954328586215228065548202, −11.25925037755360635044320686379, −10.16302585159666504076736976046, −8.851811704478158946884983000373, −7.54031646514276314576006592346, −6.11097717284170366577635046256, −5.26840326735560715748487049569, −4.48073631061681286237257232700, −2.93058766300651227263226916171, −1.13250124010902434281094106825,
1.13250124010902434281094106825, 2.93058766300651227263226916171, 4.48073631061681286237257232700, 5.26840326735560715748487049569, 6.11097717284170366577635046256, 7.54031646514276314576006592346, 8.851811704478158946884983000373, 10.16302585159666504076736976046, 11.25925037755360635044320686379, 12.04557954328586215228065548202