| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 6·7-s + 8·8-s − 10·10-s + 18·11-s − 6·13-s + 12·14-s + 16·16-s − 2·19-s − 20·20-s + 36·22-s + 26·23-s + 25·25-s − 12·26-s + 24·28-s + 32·32-s − 30·35-s − 54·37-s − 4·38-s − 40·40-s + 78·41-s + 72·44-s + 52·46-s − 86·47-s − 13·49-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 6/7·7-s + 8-s − 10-s + 1.63·11-s − 0.461·13-s + 6/7·14-s + 16-s − 0.105·19-s − 20-s + 1.63·22-s + 1.13·23-s + 25-s − 0.461·26-s + 6/7·28-s + 32-s − 6/7·35-s − 1.45·37-s − 0.105·38-s − 40-s + 1.90·41-s + 1.63·44-s + 1.13·46-s − 1.82·47-s − 0.265·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.182489901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182489901\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 6 T + p^{2} T^{2} \) |
| 11 | \( 1 - 18 T + p^{2} T^{2} \) |
| 13 | \( 1 + 6 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 2 T + p^{2} T^{2} \) |
| 23 | \( 1 - 26 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 54 T + p^{2} T^{2} \) |
| 41 | \( 1 - 78 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 86 T + p^{2} T^{2} \) |
| 53 | \( 1 + 74 T + p^{2} T^{2} \) |
| 59 | \( 1 + 78 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 18 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−11.38673238066606902562549179979, −10.85082397843840660453259077731, −9.331322961280106374094385055959, −8.192093484412768679004189813496, −7.26397228177826960934352269232, −6.44478007681662435418818637989, −5.00049694413491315299898642381, −4.25972185421254717897487242303, −3.20455338428428181019416584349, −1.46008761334736541061990965541,
1.46008761334736541061990965541, 3.20455338428428181019416584349, 4.25972185421254717897487242303, 5.00049694413491315299898642381, 6.44478007681662435418818637989, 7.26397228177826960934352269232, 8.192093484412768679004189813496, 9.331322961280106374094385055959, 10.85082397843840660453259077731, 11.38673238066606902562549179979
