| L(s) = 1 | − 2-s + 3-s + 4-s + 0.354·5-s − 6-s + 1.61·7-s − 8-s + 9-s − 0.354·10-s + 0.127·11-s + 12-s + 5.66·13-s − 1.61·14-s + 0.354·15-s + 16-s + 17-s − 18-s + 2.49·19-s + 0.354·20-s + 1.61·21-s − 0.127·22-s + 6.45·23-s − 24-s − 4.87·25-s − 5.66·26-s + 27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.158·5-s − 0.408·6-s + 0.611·7-s − 0.353·8-s + 0.333·9-s − 0.112·10-s + 0.0383·11-s + 0.288·12-s + 1.57·13-s − 0.432·14-s + 0.0916·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.573·19-s + 0.0793·20-s + 0.353·21-s − 0.0271·22-s + 1.34·23-s − 0.204·24-s − 0.974·25-s − 1.11·26-s + 0.192·27-s + 0.305·28-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
| \(L(1)\) |
\(\approx\) |
\(2.450717307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.450717307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.354T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.127T + 11T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 - 6.45T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 + 2.13T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 3.75T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 - 9.25T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.062526503356292525095103582732, −7.66307097195334888344438382737, −6.78396141138978142603308316915, −6.09110717772066993451260576908, −5.29812659060880857112504967985, −4.33848927483205545508906387511, −3.46103254804347451662881812638, −2.72445592841772857550680264735, −1.62829163196057802641827715252, −0.987792757555782904020879986209,
0.987792757555782904020879986209, 1.62829163196057802641827715252, 2.72445592841772857550680264735, 3.46103254804347451662881812638, 4.33848927483205545508906387511, 5.29812659060880857112504967985, 6.09110717772066993451260576908, 6.78396141138978142603308316915, 7.66307097195334888344438382737, 8.062526503356292525095103582732
