L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s − 10·7-s + 8·8-s + 9·9-s − 4·10-s + 10·11-s − 12·12-s − 20·14-s + 6·15-s + 16·16-s + 18·18-s − 8·20-s + 30·21-s + 20·22-s − 24·24-s − 21·25-s − 27·27-s − 40·28-s − 50·29-s + 12·30-s + 38·31-s + 32·32-s − 30·33-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 2/5·5-s − 6-s − 1.42·7-s + 8-s + 9-s − 2/5·10-s + 0.909·11-s − 12-s − 1.42·14-s + 2/5·15-s + 16-s + 18-s − 2/5·20-s + 10/7·21-s + 0.909·22-s − 24-s − 0.839·25-s − 27-s − 1.42·28-s − 1.72·29-s + 2/5·30-s + 1.22·31-s + 32-s − 0.909·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.109272074\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109272074\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
good | 5 | \( 1 + 2 T + p^{2} T^{2} \) |
| 7 | \( 1 + 10 T + p^{2} T^{2} \) |
| 11 | \( 1 - 10 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 50 T + p^{2} T^{2} \) |
| 31 | \( 1 - 38 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 94 T + p^{2} T^{2} \) |
| 59 | \( 1 - 10 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 50 T + p^{2} T^{2} \) |
| 79 | \( 1 + 58 T + p^{2} T^{2} \) |
| 83 | \( 1 + 134 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 190 T + p^{2} T^{2} \) |
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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
−17.05900125367894776227961880775, −16.19870908313102715155501224887, −15.21918399726442928042470941707, −13.44962302934968010035656911083, −12.38348407530844087816136359301, −11.41784072404285207042171459035, −9.892069267421358887642098215380, −7.04559610571669524325508573917, −5.91633467858724488813659514310, −3.91830577714857009768874565814,
3.91830577714857009768874565814, 5.91633467858724488813659514310, 7.04559610571669524325508573917, 9.892069267421358887642098215380, 11.41784072404285207042171459035, 12.38348407530844087816136359301, 13.44962302934968010035656911083, 15.21918399726442928042470941707, 16.19870908313102715155501224887, 17.05900125367894776227961880775