| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 13-s − 14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s − 21-s − 4·22-s + 8·23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.606388036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.606388036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.38234808670594, −14.19471315626779, −13.46993050015713, −13.10913970760798, −12.77956859562482, −12.32752561056152, −11.34340742983537, −11.12733501210227, −10.47465421610946, −9.893258069152100, −9.398751485313023, −8.946373309699866, −8.083883782468560, −7.704351139965842, −7.148053510066242, −6.536215667262276, −5.877055449945619, −5.339001178792485, −4.856597838130601, −4.170072457118563, −3.301694481065661, −2.955487798418658, −2.429796027484436, −1.540543425446370, −0.7296236982648874,
0.7296236982648874, 1.540543425446370, 2.429796027484436, 2.955487798418658, 3.301694481065661, 4.170072457118563, 4.856597838130601, 5.339001178792485, 5.877055449945619, 6.536215667262276, 7.148053510066242, 7.704351139965842, 8.083883782468560, 8.946373309699866, 9.398751485313023, 9.893258069152100, 10.47465421610946, 11.12733501210227, 11.34340742983537, 12.32752561056152, 12.77956859562482, 13.10913970760798, 13.46993050015713, 14.19471315626779, 14.38234808670594
