| L(s) = 1 | − 1.99·2-s − 0.450·3-s + 1.99·4-s + 2.88·5-s + 0.899·6-s − 0.362·7-s + 0.00557·8-s − 2.79·9-s − 5.76·10-s + 11-s − 0.898·12-s + 2.94·13-s + 0.725·14-s − 1.29·15-s − 4.00·16-s + 4.78·17-s + 5.59·18-s + 5.95·19-s + 5.76·20-s + 0.163·21-s − 1.99·22-s − 6.31·23-s − 0.00250·24-s + 3.32·25-s − 5.89·26-s + 2.60·27-s − 0.724·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.259·3-s + 0.998·4-s + 1.29·5-s + 0.367·6-s − 0.137·7-s + 0.00197·8-s − 0.932·9-s − 1.82·10-s + 0.301·11-s − 0.259·12-s + 0.818·13-s + 0.193·14-s − 0.335·15-s − 1.00·16-s + 1.16·17-s + 1.31·18-s + 1.36·19-s + 1.28·20-s + 0.0356·21-s − 0.426·22-s − 1.31·23-s − 0.000512·24-s + 0.664·25-s − 1.15·26-s + 0.502·27-s − 0.137·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 + 0.450T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 + 0.362T + 7T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 + 6.25T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 1.54T + 59T^{2} \) |
| 61 | \( 1 + 7.30T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 2.67T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−7.894932143389611147628676752463, −7.17201740744731047440838608585, −6.34977182035836809031989216667, −5.62001850955446913597993293054, −5.34049548388052747955138804526, −3.84969491216563681984715694587, −2.97782085553094750458447573577, −1.83731199510747064654957273119, −1.31615806313589407122361414602, 0,
1.31615806313589407122361414602, 1.83731199510747064654957273119, 2.97782085553094750458447573577, 3.84969491216563681984715694587, 5.34049548388052747955138804526, 5.62001850955446913597993293054, 6.34977182035836809031989216667, 7.17201740744731047440838608585, 7.894932143389611147628676752463
