| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 15-s + 21-s + 25-s − 27-s + 35-s + 2·41-s + 2·43-s − 45-s + 2·47-s + 49-s − 63-s − 2·67-s − 75-s + 81-s + 2·83-s − 2·89-s + 2·101-s − 105-s − 2·109-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 15-s + 21-s + 25-s − 27-s + 35-s + 2·41-s + 2·43-s − 45-s + 2·47-s + 49-s − 63-s − 2·67-s − 75-s + 81-s + 2·83-s − 2·89-s + 2·101-s − 105-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5562479403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5562479403\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.534687063562448987304743251033, −8.907491872153772077335069018681, −7.62782982554834173690099447436, −7.23310734704589149797844118640, −6.25350305436473318183415186187, −5.63572069302790546415977998565, −4.44141947574933019001486905973, −3.87415732318666004601090760942, −2.67451298884071575770590540980, −0.795795216446716496043311403187,
0.795795216446716496043311403187, 2.67451298884071575770590540980, 3.87415732318666004601090760942, 4.44141947574933019001486905973, 5.63572069302790546415977998565, 6.25350305436473318183415186187, 7.23310734704589149797844118640, 7.62782982554834173690099447436, 8.907491872153772077335069018681, 9.534687063562448987304743251033
