| L(s) = 1 | + 2.13·2-s − 3-s + 2.56·4-s + 0.126·5-s − 2.13·6-s + 7-s + 1.19·8-s + 9-s + 0.269·10-s + 5.90·11-s − 2.56·12-s − 5.46·13-s + 2.13·14-s − 0.126·15-s − 2.56·16-s + 4.40·17-s + 2.13·18-s − 1.95·19-s + 0.322·20-s − 21-s + 12.6·22-s − 9.04·23-s − 1.19·24-s − 4.98·25-s − 11.6·26-s − 27-s + 2.56·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.28·4-s + 0.0563·5-s − 0.871·6-s + 0.377·7-s + 0.423·8-s + 0.333·9-s + 0.0851·10-s + 1.78·11-s − 0.739·12-s − 1.51·13-s + 0.570·14-s − 0.0325·15-s − 0.640·16-s + 1.06·17-s + 0.503·18-s − 0.448·19-s + 0.0721·20-s − 0.218·21-s + 2.69·22-s − 1.88·23-s − 0.244·24-s − 0.996·25-s − 2.28·26-s − 0.192·27-s + 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 - 0.126T + 5T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 9.04T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 + 2.28T + 59T^{2} \) |
| 61 | \( 1 - 8.81T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 - 0.451T + 71T^{2} \) |
| 73 | \( 1 - 5.28T + 73T^{2} \) |
| 79 | \( 1 + 17.7T + 79T^{2} \) |
| 83 | \( 1 + 3.40T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.29746792204348840185547059124, −6.51608090149754929726931498827, −5.80547508251623630071664487888, −5.51389846331704032020208375969, −4.57737724868812816319489768861, −3.99369287919836559104163103265, −3.56534837068108856920013810327, −2.25415551793624837884763294410, −1.64813019788129115088872350543, 0,
1.64813019788129115088872350543, 2.25415551793624837884763294410, 3.56534837068108856920013810327, 3.99369287919836559104163103265, 4.57737724868812816319489768861, 5.51389846331704032020208375969, 5.80547508251623630071664487888, 6.51608090149754929726931498827, 7.29746792204348840185547059124
