| L(s) = 1 | − 0.857·2-s − 3-s − 1.26·4-s − 4.41·5-s + 0.857·6-s + 0.00991·7-s + 2.79·8-s + 9-s + 3.78·10-s − 3.26·11-s + 1.26·12-s + 2.70·13-s − 0.00850·14-s + 4.41·15-s + 0.126·16-s − 17-s − 0.857·18-s − 4.27·19-s + 5.58·20-s − 0.00991·21-s + 2.80·22-s − 8.57·23-s − 2.79·24-s + 14.5·25-s − 2.31·26-s − 27-s − 0.0125·28-s + ⋯ |
| L(s) = 1 | − 0.606·2-s − 0.577·3-s − 0.632·4-s − 1.97·5-s + 0.350·6-s + 0.00374·7-s + 0.989·8-s + 0.333·9-s + 1.19·10-s − 0.984·11-s + 0.364·12-s + 0.749·13-s − 0.00227·14-s + 1.14·15-s + 0.0316·16-s − 0.242·17-s − 0.202·18-s − 0.980·19-s + 1.24·20-s − 0.00216·21-s + 0.597·22-s − 1.78·23-s − 0.571·24-s + 2.90·25-s − 0.454·26-s − 0.192·27-s − 0.00236·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.005658811546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005658811546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.857T + 2T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 - 0.00991T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 - 0.575T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 6.91T + 73T^{2} \) |
| 83 | \( 1 - 0.651T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.367876641615097576738676298289, −7.903322144005323976785851536902, −7.24791846590248825384745494316, −6.38396740655429920235470735222, −5.30726334016021227745101574566, −4.49778737315380567411703724714, −4.04822430573724579838247397414, −3.19842422126770713752897105664, −1.57481477815035479204691276279, −0.04968889666658495381935536458,
0.04968889666658495381935536458, 1.57481477815035479204691276279, 3.19842422126770713752897105664, 4.04822430573724579838247397414, 4.49778737315380567411703724714, 5.30726334016021227745101574566, 6.38396740655429920235470735222, 7.24791846590248825384745494316, 7.903322144005323976785851536902, 8.367876641615097576738676298289
