L(s) = 1 | + 27·3-s + 64·4-s + 729·9-s + 1.72e3·12-s − 506·13-s + 4.09e3·16-s + 1.05e4·19-s + 1.56e4·25-s + 1.96e4·27-s − 3.52e4·31-s + 4.66e4·36-s − 8.92e4·37-s − 1.36e4·39-s + 1.11e5·43-s + 1.10e5·48-s − 3.23e4·52-s + 2.85e5·57-s + 4.20e5·61-s + 2.62e5·64-s + 1.72e5·67-s − 6.38e5·73-s + 4.21e5·75-s + 6.77e5·76-s − 2.04e5·79-s + 5.31e5·81-s − 9.52e5·93-s + 5.64e4·97-s + ⋯ |
L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 0.230·13-s + 16-s + 1.54·19-s + 25-s + 27-s − 1.18·31-s + 36-s − 1.76·37-s − 0.230·39-s + 1.40·43-s + 48-s − 0.230·52-s + 1.54·57-s + 1.85·61-s + 64-s + 0.574·67-s − 1.64·73-s + 75-s + 1.54·76-s − 0.415·79-s + 81-s − 1.18·93-s + 0.0618·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.971880789\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.971880789\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p^{3} T \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 + 506 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 - 10582 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 + 35282 T + p^{6} T^{2} \) |
| 37 | \( 1 + 89206 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 111386 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 420838 T + p^{6} T^{2} \) |
| 67 | \( 1 - 172874 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 + 638066 T + p^{6} T^{2} \) |
| 79 | \( 1 + 204622 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 - 56446 T + p^{6} T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.00473331040157158361453171310, −10.85086782452769874522246813102, −9.863184929251031372496498723432, −8.782707000171797425573290376172, −7.56665393000645440993927343623, −6.91861484381933415715502241166, −5.34485273813014811222223112889, −3.61076593966672307252424922423, −2.58371560799500511905402004407, −1.31652732915373072280305900170,
1.31652732915373072280305900170, 2.58371560799500511905402004407, 3.61076593966672307252424922423, 5.34485273813014811222223112889, 6.91861484381933415715502241166, 7.56665393000645440993927343623, 8.782707000171797425573290376172, 9.863184929251031372496498723432, 10.85086782452769874522246813102, 12.00473331040157158361453171310