L(s) = 1 | + 2-s − 3-s + 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 13-s + 14-s − 15-s − 16-s + 18-s − 19-s − 21-s − 22-s + 24-s − 26-s − 27-s + 29-s − 30-s + 33-s + 35-s − 37-s − 38-s + 39-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 13-s + 14-s − 15-s − 16-s + 18-s − 19-s − 21-s − 22-s + 24-s − 26-s − 27-s + 29-s − 30-s + 33-s + 35-s − 37-s − 38-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9001109165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9001109165\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.52774274759651952795141339305, −11.73652704061383165115573225475, −10.60078208166062937669383107989, −9.869212863884393600082192228144, −8.497126690422111619558314566805, −7.05474688251435185853047927074, −5.82109089115999547293789376839, −5.18815546221459090870871092600, −4.38576489093358455968417944071, −2.30109140277429904126234426023,
2.30109140277429904126234426023, 4.38576489093358455968417944071, 5.18815546221459090870871092600, 5.82109089115999547293789376839, 7.05474688251435185853047927074, 8.497126690422111619558314566805, 9.869212863884393600082192228144, 10.60078208166062937669383107989, 11.73652704061383165115573225475, 12.52774274759651952795141339305