L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s − 15-s + 16-s − 18-s + 20-s − 22-s + 24-s − 27-s + 29-s + 30-s − 31-s − 32-s − 33-s + 36-s − 40-s + 44-s + 45-s − 48-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s − 15-s + 16-s − 18-s + 20-s − 22-s + 24-s − 27-s + 29-s + 30-s − 31-s − 32-s − 33-s + 36-s − 40-s + 44-s + 45-s − 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6425046856\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6425046856\) |
\(L(1)\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.912736823186251069305112354368, −9.387382770402993747373823412182, −8.518603449405621620195511856016, −7.35100983172926698446391031517, −6.61081794820260449350823538573, −6.01754372167287612008607827805, −5.18647715230684529380301050857, −3.80239448679299552640242337277, −2.21974967604645260546398307986, −1.18516225414688112075824373034,
1.18516225414688112075824373034, 2.21974967604645260546398307986, 3.80239448679299552640242337277, 5.18647715230684529380301050857, 6.01754372167287612008607827805, 6.61081794820260449350823538573, 7.35100983172926698446391031517, 8.518603449405621620195511856016, 9.387382770402993747373823412182, 9.912736823186251069305112354368