| L(s) = 1 | − 3.54·2-s − 2.15·3-s + 4.54·4-s + 6.95·5-s + 7.63·6-s + 7·7-s + 12.2·8-s − 22.3·9-s − 24.6·10-s − 54.7·11-s − 9.80·12-s + 8.53·13-s − 24.7·14-s − 14.9·15-s − 79.6·16-s + 62.6·17-s + 79.1·18-s + 88.4·19-s + 31.6·20-s − 15.0·21-s + 193.·22-s + 23·23-s − 26.3·24-s − 76.6·25-s − 30.2·26-s + 106.·27-s + 31.8·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.414·3-s + 0.568·4-s + 0.622·5-s + 0.519·6-s + 0.377·7-s + 0.540·8-s − 0.827·9-s − 0.779·10-s − 1.49·11-s − 0.235·12-s + 0.182·13-s − 0.473·14-s − 0.258·15-s − 1.24·16-s + 0.893·17-s + 1.03·18-s + 1.06·19-s + 0.353·20-s − 0.156·21-s + 1.87·22-s + 0.208·23-s − 0.224·24-s − 0.612·25-s − 0.228·26-s + 0.758·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7177207171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7177207171\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 3.54T + 8T^{2} \) |
| 3 | \( 1 + 2.15T + 27T^{2} \) |
| 5 | \( 1 - 6.95T + 125T^{2} \) |
| 11 | \( 1 + 54.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.53T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 450.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 522.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 317.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 699.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 225.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 415.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 18.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 945.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 795.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−12.14177264735223432443597377729, −11.03828161773049071470213145309, −10.29354388094523544206508278054, −9.447169916596136662269554278984, −8.259544495689443186379034401583, −7.59127063114807033611605478235, −5.95449043409692466660319037341, −4.97101526763009422107657436266, −2.62134566953475994401504504838, −0.844548971227182311482623532936,
0.844548971227182311482623532936, 2.62134566953475994401504504838, 4.97101526763009422107657436266, 5.95449043409692466660319037341, 7.59127063114807033611605478235, 8.259544495689443186379034401583, 9.447169916596136662269554278984, 10.29354388094523544206508278054, 11.03828161773049071470213145309, 12.14177264735223432443597377729
