| L(s) = 1 | − 1.62·2-s + 2.31·3-s + 0.652·4-s + 5-s − 3.76·6-s − 4.05·7-s + 2.19·8-s + 2.35·9-s − 1.62·10-s − 0.153·11-s + 1.51·12-s − 0.775·13-s + 6.60·14-s + 2.31·15-s − 4.87·16-s − 17-s − 3.83·18-s + 5.92·19-s + 0.652·20-s − 9.38·21-s + 0.249·22-s + 5.92·23-s + 5.07·24-s + 25-s + 1.26·26-s − 1.49·27-s − 2.64·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 1.33·3-s + 0.326·4-s + 0.447·5-s − 1.53·6-s − 1.53·7-s + 0.775·8-s + 0.784·9-s − 0.515·10-s − 0.0462·11-s + 0.435·12-s − 0.215·13-s + 1.76·14-s + 0.597·15-s − 1.21·16-s − 0.242·17-s − 0.903·18-s + 1.35·19-s + 0.145·20-s − 2.04·21-s + 0.0532·22-s + 1.23·23-s + 1.03·24-s + 0.200·25-s + 0.247·26-s − 0.287·27-s − 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 0.153T + 11T^{2} \) |
| 13 | \( 1 + 0.775T + 13T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 - 3.58T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 - 7.77T + 97T^{2} \) |
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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
−7.81454273436815064087476143346, −7.27227250166479658559881105683, −6.71146546323014466059342143890, −5.69740506361423425867063915291, −4.78207238948292231083707143769, −3.59527431301326792440892290555, −3.13350886748603715093691938390, −2.28719265724217710964664459974, −1.30210964424831399013370930313, 0,
1.30210964424831399013370930313, 2.28719265724217710964664459974, 3.13350886748603715093691938390, 3.59527431301326792440892290555, 4.78207238948292231083707143769, 5.69740506361423425867063915291, 6.71146546323014466059342143890, 7.27227250166479658559881105683, 7.81454273436815064087476143346
