| L(s) = 1 | − 1.65·2-s − 1.14·3-s + 0.734·4-s − 5-s + 1.88·6-s + 7-s + 2.09·8-s − 1.69·9-s + 1.65·10-s − 0.838·12-s + 5.60·13-s − 1.65·14-s + 1.14·15-s − 4.92·16-s − 4.57·17-s + 2.80·18-s + 1.88·19-s − 0.734·20-s − 1.14·21-s + 2.08·23-s − 2.38·24-s + 25-s − 9.26·26-s + 5.36·27-s + 0.734·28-s − 5.51·29-s − 1.88·30-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.659·3-s + 0.367·4-s − 0.447·5-s + 0.770·6-s + 0.377·7-s + 0.739·8-s − 0.565·9-s + 0.522·10-s − 0.242·12-s + 1.55·13-s − 0.441·14-s + 0.294·15-s − 1.23·16-s − 1.11·17-s + 0.661·18-s + 0.431·19-s − 0.164·20-s − 0.249·21-s + 0.433·23-s − 0.487·24-s + 0.200·25-s − 1.81·26-s + 1.03·27-s + 0.138·28-s − 1.02·29-s − 0.344·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 - 0.696T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 - 5.93T + 97T^{2} \) |
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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
−8.266738882205082784186440670241, −7.39414972554395377339443733384, −6.79665000311362658640308343577, −5.87090610975042762074685331209, −5.16513244309112556069637408000, −4.25206626180890219927236749433, −3.42583463877008724542717314681, −2.04610345758136170996818903819, −1.03918884422847804902374179028, 0,
1.03918884422847804902374179028, 2.04610345758136170996818903819, 3.42583463877008724542717314681, 4.25206626180890219927236749433, 5.16513244309112556069637408000, 5.87090610975042762074685331209, 6.79665000311362658640308343577, 7.39414972554395377339443733384, 8.266738882205082784186440670241
