| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 50·5-s − 36·6-s + 2·7-s + 64·8-s + 81·9-s + 200·10-s − 121·11-s − 144·12-s + 966·13-s + 8·14-s − 450·15-s + 256·16-s + 1.96e3·17-s + 324·18-s + 1.24e3·19-s + 800·20-s − 18·21-s − 484·22-s + 136·23-s − 576·24-s − 625·25-s + 3.86e3·26-s − 729·27-s + 32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.0154·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.58·13-s + 0.0109·14-s − 0.516·15-s + 1/4·16-s + 1.64·17-s + 0.235·18-s + 0.791·19-s + 0.447·20-s − 0.00890·21-s − 0.213·22-s + 0.0536·23-s − 0.204·24-s − 1/5·25-s + 1.12·26-s − 0.192·27-s + 0.00771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.698983138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.698983138\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 2 p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 - 2 T + p^{5} T^{2} \) |
| 13 | \( 1 - 966 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1964 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1246 T + p^{5} T^{2} \) |
| 23 | \( 1 - 136 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7824 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4752 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4650 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7536 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14582 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3984 T + p^{5} T^{2} \) |
| 53 | \( 1 - 12350 T + p^{5} T^{2} \) |
| 59 | \( 1 + 22380 T + p^{5} T^{2} \) |
| 61 | \( 1 + 15662 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29564 T + p^{5} T^{2} \) |
| 71 | \( 1 - 55536 T + p^{5} T^{2} \) |
| 73 | \( 1 + 63258 T + p^{5} T^{2} \) |
| 79 | \( 1 + 514 p T + p^{5} T^{2} \) |
| 83 | \( 1 - 81808 T + p^{5} T^{2} \) |
| 89 | \( 1 - 116434 T + p^{5} T^{2} \) |
| 97 | \( 1 - 20734 T + p^{5} T^{2} \) |
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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
−13.67407134878425170614608412223, −12.94347276495758191486878572733, −11.67652992030430118616587486161, −10.65526608401205724352077570706, −9.482836328352781141255523839964, −7.66023411908080798959607871776, −6.07115407740526164410514851844, −5.40500645041895651059846523723, −3.53096331264009716799644160689, −1.46958067397625801295955193971,
1.46958067397625801295955193971, 3.53096331264009716799644160689, 5.40500645041895651059846523723, 6.07115407740526164410514851844, 7.66023411908080798959607871776, 9.482836328352781141255523839964, 10.65526608401205724352077570706, 11.67652992030430118616587486161, 12.94347276495758191486878572733, 13.67407134878425170614608412223
