| L(s) = 1 | + 2-s − 2.96·3-s + 4-s + 3.59·5-s − 2.96·6-s − 0.0161·7-s + 8-s + 5.81·9-s + 3.59·10-s − 0.643·11-s − 2.96·12-s − 0.958·13-s − 0.0161·14-s − 10.6·15-s + 16-s − 0.421·17-s + 5.81·18-s − 5.37·19-s + 3.59·20-s + 0.0480·21-s − 0.643·22-s − 5.41·23-s − 2.96·24-s + 7.89·25-s − 0.958·26-s − 8.37·27-s − 0.0161·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s + 1.60·5-s − 1.21·6-s − 0.00611·7-s + 0.353·8-s + 1.93·9-s + 1.13·10-s − 0.194·11-s − 0.857·12-s − 0.265·13-s − 0.00432·14-s − 2.75·15-s + 0.250·16-s − 0.102·17-s + 1.37·18-s − 1.23·19-s + 0.802·20-s + 0.0104·21-s − 0.137·22-s − 1.12·23-s − 0.606·24-s + 1.57·25-s − 0.187·26-s − 1.61·27-s − 0.00305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 + 0.0161T + 7T^{2} \) |
| 11 | \( 1 + 0.643T + 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 + 0.421T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 8.56T + 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 + 0.270T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + 9.01T + 53T^{2} \) |
| 59 | \( 1 + 2.03T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 - 0.119T + 73T^{2} \) |
| 79 | \( 1 + 0.556T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.81608292027722843290355025074, −6.81119151896627282221193159340, −6.35773119878042797959867978006, −5.81516620851996416699616774950, −5.23549068183422285673099071934, −4.67954122800988416463102261486, −3.63505680331187875967980196504, −2.16269275314805474489473160710, −1.63602216479313282256771834612, 0,
1.63602216479313282256771834612, 2.16269275314805474489473160710, 3.63505680331187875967980196504, 4.67954122800988416463102261486, 5.23549068183422285673099071934, 5.81516620851996416699616774950, 6.35773119878042797959867978006, 6.81119151896627282221193159340, 7.81608292027722843290355025074
