L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 12-s − 13-s − 16-s + 18-s − 6·19-s + 8·23-s + 3·24-s − 26-s − 27-s + 2·29-s + 10·31-s + 5·32-s − 36-s − 8·37-s − 6·38-s + 39-s − 10·41-s + 6·43-s + 8·46-s − 2·47-s + 48-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.235·18-s − 1.37·19-s + 1.66·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.883·32-s − 1/6·36-s − 1.31·37-s − 0.973·38-s + 0.160·39-s − 1.56·41-s + 0.914·43-s + 1.17·46-s − 0.291·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.345224116\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345224116\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.55018708672238, −13.93477805458679, −13.58409612768473, −12.98440221502226, −12.48729150055658, −12.27073050127338, −11.53671821190010, −11.09426015294366, −10.37683182859565, −10.07157834572219, −9.308960729650611, −8.783704755276007, −8.378763092641210, −7.656862600392631, −6.822998243128948, −6.466646258529203, −5.968356987685388, −5.126222485957331, −4.809238110423281, −4.418313347436295, −3.550227123740003, −3.052242403259947, −2.264309306598838, −1.281412074546049, −0.4049003798414124,
0.4049003798414124, 1.281412074546049, 2.264309306598838, 3.052242403259947, 3.550227123740003, 4.418313347436295, 4.809238110423281, 5.126222485957331, 5.968356987685388, 6.466646258529203, 6.822998243128948, 7.656862600392631, 8.378763092641210, 8.783704755276007, 9.308960729650611, 10.07157834572219, 10.37683182859565, 11.09426015294366, 11.53671821190010, 12.27073050127338, 12.48729150055658, 12.98440221502226, 13.58409612768473, 13.93477805458679, 14.55018708672238