| L(s) = 1 | − 2-s + 3-s + 4-s + 1.98·5-s − 6-s + 3.61·7-s − 8-s + 9-s − 1.98·10-s + 5.33·11-s + 12-s + 13-s − 3.61·14-s + 1.98·15-s + 16-s + 2.87·17-s − 18-s − 2.97·19-s + 1.98·20-s + 3.61·21-s − 5.33·22-s − 3.89·23-s − 24-s − 1.06·25-s − 26-s + 27-s + 3.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.887·5-s − 0.408·6-s + 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.627·10-s + 1.60·11-s + 0.288·12-s + 0.277·13-s − 0.967·14-s + 0.512·15-s + 0.250·16-s + 0.697·17-s − 0.235·18-s − 0.683·19-s + 0.443·20-s + 0.789·21-s − 1.13·22-s − 0.811·23-s − 0.204·24-s − 0.212·25-s − 0.196·26-s + 0.192·27-s + 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.247163309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.247163309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 1.98T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 - 0.527T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + 0.778T + 41T^{2} \) |
| 43 | \( 1 + 6.00T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 + 8.42T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.20T + 97T^{2} \) |
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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
−8.034645377744071076115273351613, −7.33998713724736974834105961694, −6.36177739236902008411580550542, −6.08066592034669433860010384167, −5.00649630771123289717900177998, −4.23870898475024576082087260519, −3.45305984230499499282838712548, −2.28091314964928318166343079331, −1.70402755759408013457804936502, −1.08238281958044933475655953299,
1.08238281958044933475655953299, 1.70402755759408013457804936502, 2.28091314964928318166343079331, 3.45305984230499499282838712548, 4.23870898475024576082087260519, 5.00649630771123289717900177998, 6.08066592034669433860010384167, 6.36177739236902008411580550542, 7.33998713724736974834105961694, 8.034645377744071076115273351613
