| L(s) = 1 | − 2-s + 3-s + 4-s + 0.556·5-s − 6-s + 0.727·7-s − 8-s + 9-s − 0.556·10-s + 4.53·11-s + 12-s − 2.52·13-s − 0.727·14-s + 0.556·15-s + 16-s − 17-s − 18-s − 7.07·19-s + 0.556·20-s + 0.727·21-s − 4.53·22-s − 0.184·23-s − 24-s − 4.68·25-s + 2.52·26-s + 27-s + 0.727·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.249·5-s − 0.408·6-s + 0.274·7-s − 0.353·8-s + 0.333·9-s − 0.176·10-s + 1.36·11-s + 0.288·12-s − 0.699·13-s − 0.194·14-s + 0.143·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.62·19-s + 0.124·20-s + 0.158·21-s − 0.965·22-s − 0.0385·23-s − 0.204·24-s − 0.937·25-s + 0.494·26-s + 0.192·27-s + 0.137·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 0.556T + 5T^{2} \) |
| 7 | \( 1 - 0.727T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 0.184T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−7.80541530543327157770899813815, −7.16803338169669176183503305785, −6.41356143753825623728581199634, −5.86952755988410634491977770756, −4.61941922550774442835606652366, −4.04957994362007286572383417615, −3.07135416984507307080157225759, −2.03572145844700864852868367929, −1.55929368621610468772230904596, 0,
1.55929368621610468772230904596, 2.03572145844700864852868367929, 3.07135416984507307080157225759, 4.04957994362007286572383417615, 4.61941922550774442835606652366, 5.86952755988410634491977770756, 6.41356143753825623728581199634, 7.16803338169669176183503305785, 7.80541530543327157770899813815
