L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 3·11-s − 4·13-s + 15-s + 4·19-s − 4·21-s − 6·23-s + 25-s − 27-s + 3·29-s + 3·33-s − 4·35-s − 4·37-s + 4·39-s − 9·41-s + 10·43-s − 45-s + 12·47-s + 9·49-s + 6·53-s + 3·55-s − 4·57-s − 3·59-s − 7·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.522·33-s − 0.676·35-s − 0.657·37-s + 0.640·39-s − 1.40·41-s + 1.52·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.404·55-s − 0.529·57-s − 0.390·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.613847961\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613847961\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.63098957100954, −12.29686507412181, −12.08713095744209, −11.50773634125980, −11.18718396799612, −10.56123476321222, −10.31006808050581, −9.829180885292688, −9.125963853407488, −8.657351818136735, −7.983245737860193, −7.728985233694858, −7.429953528529538, −6.832800578498061, −6.161062088347322, −5.491657773717195, −5.098295079402052, −4.923782907889330, −4.152869436873705, −3.821977080865768, −2.873883942539477, −2.348649801206711, −1.829827445108324, −1.053557538925367, −0.4047478995993170,
0.4047478995993170, 1.053557538925367, 1.829827445108324, 2.348649801206711, 2.873883942539477, 3.821977080865768, 4.152869436873705, 4.923782907889330, 5.098295079402052, 5.491657773717195, 6.161062088347322, 6.832800578498061, 7.429953528529538, 7.728985233694858, 7.983245737860193, 8.657351818136735, 9.125963853407488, 9.829180885292688, 10.31006808050581, 10.56123476321222, 11.18718396799612, 11.50773634125980, 12.08713095744209, 12.29686507412181, 12.63098957100954