| L(s) = 1 | − 2.31·2-s − 3·3-s − 2.65·4-s − 6.33·5-s + 6.93·6-s + 17.5·7-s + 24.6·8-s + 9·9-s + 14.6·10-s − 16.1·11-s + 7.96·12-s + 71.6·13-s − 40.6·14-s + 18.9·15-s − 35.7·16-s + 14.0·17-s − 20.8·18-s + 129.·19-s + 16.8·20-s − 52.7·21-s + 37.4·22-s − 23·23-s − 73.9·24-s − 84.9·25-s − 165.·26-s − 27·27-s − 46.6·28-s + ⋯ |
| L(s) = 1 | − 0.817·2-s − 0.577·3-s − 0.331·4-s − 0.566·5-s + 0.471·6-s + 0.950·7-s + 1.08·8-s + 0.333·9-s + 0.462·10-s − 0.443·11-s + 0.191·12-s + 1.52·13-s − 0.776·14-s + 0.326·15-s − 0.558·16-s + 0.200·17-s − 0.272·18-s + 1.55·19-s + 0.187·20-s − 0.548·21-s + 0.362·22-s − 0.208·23-s − 0.628·24-s − 0.679·25-s − 1.24·26-s − 0.192·27-s − 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7248805235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7248805235\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.31T + 8T^{2} \) |
| 5 | \( 1 + 6.33T + 125T^{2} \) |
| 7 | \( 1 - 17.5T + 343T^{2} \) |
| 11 | \( 1 + 16.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 70.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 93.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 389.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 87.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 341.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 612.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 226.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 678.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 45.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 585.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−14.17199214077277136150773183141, −13.16675463472976862191658435173, −11.61427895997599934257966618603, −10.93652124483429389188108437476, −9.667572508112709288014549918395, −8.312472713994209862476576298500, −7.54929994408598862079196220939, −5.57268272022412019142522176134, −4.14127477592261345809332921604, −1.05066521813749114512041956298,
1.05066521813749114512041956298, 4.14127477592261345809332921604, 5.57268272022412019142522176134, 7.54929994408598862079196220939, 8.312472713994209862476576298500, 9.667572508112709288014549918395, 10.93652124483429389188108437476, 11.61427895997599934257966618603, 13.16675463472976862191658435173, 14.17199214077277136150773183141
