L(s) = 1 | + (−0.780 + 1.17i)2-s + (−0.780 − 1.84i)4-s − 1.69i·5-s + (−2.56 + 0.662i)7-s + (2.78 + 0.516i)8-s + (2 + 1.32i)10-s − 3.02i·11-s − 6.04i·13-s + (1.21 − 3.53i)14-s + (−2.78 + 2.87i)16-s − 4.34i·17-s + 1.12·19-s + (−3.12 + 1.32i)20-s + (3.56 + 2.35i)22-s + 3.02i·23-s + ⋯ |
L(s) = 1 | + (−0.552 + 0.833i)2-s + (−0.390 − 0.920i)4-s − 0.758i·5-s + (−0.968 + 0.250i)7-s + (0.983 + 0.182i)8-s + (0.632 + 0.418i)10-s − 0.910i·11-s − 1.67i·13-s + (0.325 − 0.945i)14-s + (−0.695 + 0.718i)16-s − 1.05i·17-s + 0.257·19-s + (−0.698 + 0.296i)20-s + (0.759 + 0.502i)22-s + 0.629i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.623632 - 0.307729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.623632 - 0.307729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.780 - 1.17i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.56 - 0.662i)T \) |
good | 5 | \( 1 + 1.69iT - 5T^{2} \) |
| 11 | \( 1 + 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.04iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 3.02iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 7.73iT - 41T^{2} \) |
| 43 | \( 1 + 8.10iT - 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 9.43iT - 61T^{2} \) |
| 67 | \( 1 - 2.06iT - 67T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 - 4.71iT - 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−11.95591648484610419118707777558, −10.63500972691350806749014516502, −9.767422053145232604109193081730, −8.884323767157839161638134140045, −8.112271176030091713646032641211, −6.96874588769461866320379606334, −5.78057953225196759515243928198, −5.08843144426301272963333119371, −3.20508308914145837452820012099, −0.66884033543823757728129316142,
2.00998914018690913983320959374, 3.39538406667254509395169331491, 4.46711452391305228634967994700, 6.57083924130414393763449692854, 7.14431026890047472265958664964, 8.561676929671817800086832661022, 9.559191489201107256203160067156, 10.24009029761968135095983801344, 11.08954943750795433849530752053, 12.13043002636508401933059666395