L(s) = 1 | + 5.27·2-s − 9·3-s − 4.20·4-s − 17.5·5-s − 47.4·6-s + 172.·7-s − 190.·8-s + 81·9-s − 92.4·10-s + 249.·11-s + 37.8·12-s + 282.·13-s + 910.·14-s + 157.·15-s − 871.·16-s − 1.34e3·17-s + 427.·18-s − 2.88e3·19-s + 73.7·20-s − 1.55e3·21-s + 1.31e3·22-s + 692.·23-s + 1.71e3·24-s − 2.81e3·25-s + 1.49e3·26-s − 729·27-s − 726.·28-s + ⋯ |
L(s) = 1 | + 0.931·2-s − 0.577·3-s − 0.131·4-s − 0.313·5-s − 0.538·6-s + 1.33·7-s − 1.05·8-s + 0.333·9-s − 0.292·10-s + 0.621·11-s + 0.0758·12-s + 0.464·13-s + 1.24·14-s + 0.181·15-s − 0.851·16-s − 1.13·17-s + 0.310·18-s − 1.83·19-s + 0.0412·20-s − 0.769·21-s + 0.579·22-s + 0.273·23-s + 0.608·24-s − 0.901·25-s + 0.432·26-s − 0.192·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
good | 2 | \( 1 - 5.27T + 32T^{2} \) |
| 5 | \( 1 + 17.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 249.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 282.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 692.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 828.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.79e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.94e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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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
−11.39637369240011798106982737023, −10.81474807539144533494645917105, −9.117404203595384869698217638736, −8.270498837026490629677493230863, −6.77093323610824169783531742558, −5.69685737904594283415334147497, −4.58267620405430404759562377617, −3.92979710507895581360082710614, −1.87190997829875665314687028645, 0,
1.87190997829875665314687028645, 3.92979710507895581360082710614, 4.58267620405430404759562377617, 5.69685737904594283415334147497, 6.77093323610824169783531742558, 8.270498837026490629677493230863, 9.117404203595384869698217638736, 10.81474807539144533494645917105, 11.39637369240011798106982737023