| L(s) = 1 | + 2-s − 3-s + 4-s + 1.63·5-s − 6-s + 7-s + 8-s + 9-s + 1.63·10-s + 2.56·11-s − 12-s + 0.379·13-s + 14-s − 1.63·15-s + 16-s + 1.75·17-s + 18-s + 4.45·19-s + 1.63·20-s − 21-s + 2.56·22-s + 1.61·23-s − 24-s − 2.33·25-s + 0.379·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.729·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.516·10-s + 0.773·11-s − 0.288·12-s + 0.105·13-s + 0.267·14-s − 0.421·15-s + 0.250·16-s + 0.425·17-s + 0.235·18-s + 1.02·19-s + 0.364·20-s − 0.218·21-s + 0.546·22-s + 0.337·23-s − 0.204·24-s − 0.467·25-s + 0.0743·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.871977806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.871977806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.379T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 - 4.45T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 + 6.49T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 + 5.03T + 97T^{2} \) |
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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
−7.52441455663468988261431175023, −7.07674517173138471393850576125, −6.22530955782016941384524756036, −5.61080538223670638965684402096, −5.29916793529324147336101822302, −4.27295002258681667300177433737, −3.72577070792896420362900483985, −2.68125097252086949802050804919, −1.75869682044288954972415157732, −0.967369020715271251186657375136,
0.967369020715271251186657375136, 1.75869682044288954972415157732, 2.68125097252086949802050804919, 3.72577070792896420362900483985, 4.27295002258681667300177433737, 5.29916793529324147336101822302, 5.61080538223670638965684402096, 6.22530955782016941384524756036, 7.07674517173138471393850576125, 7.52441455663468988261431175023
