L(s) = 1 | + (1.60 − 0.169i)2-s + (1.58 − 0.336i)4-s + (−0.994 − 0.104i)5-s + (0.809 − 1.40i)7-s + (0.951 − 0.309i)8-s − 1.61·10-s + (0.994 − 0.104i)11-s + (1.06 − 2.39i)14-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−1.60 + 0.169i)20-s + (1.58 − 0.336i)22-s + (−0.406 + 0.913i)23-s + (0.978 + 0.207i)25-s + (0.809 − 2.48i)28-s + (0.459 − 0.413i)29-s + ⋯ |
L(s) = 1 | + (1.60 − 0.169i)2-s + (1.58 − 0.336i)4-s + (−0.994 − 0.104i)5-s + (0.809 − 1.40i)7-s + (0.951 − 0.309i)8-s − 1.61·10-s + (0.994 − 0.104i)11-s + (1.06 − 2.39i)14-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−1.60 + 0.169i)20-s + (1.58 − 0.336i)22-s + (−0.406 + 0.913i)23-s + (0.978 + 0.207i)25-s + (0.809 − 2.48i)28-s + (0.459 − 0.413i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.587843560\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.587843560\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.994 + 0.104i)T \) |
good | 2 | \( 1 + (-1.60 + 0.169i)T + (0.978 - 0.207i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \) |
| 13 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.406 - 0.913i)T + (-0.669 - 0.743i)T^{2} \) |
| 29 | \( 1 + (-0.459 + 0.413i)T + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \) |
| 71 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.413 + 0.459i)T + (-0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
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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
−9.122594715317702386601875892147, −8.279435450925254325211716585144, −7.30721857289090903981208710229, −6.86883926554081522631385750259, −5.92260130639549746495088576482, −4.75510169690889102758477908963, −4.19937552282842183031116949491, −3.91583368966335895300921690906, −2.72043120005091552911905835104, −1.28176069908007638600874702884,
1.99696746545055075477370263899, 2.88130654237421417709723099510, 3.97912438167876309891630235108, 4.48051857572089131463030349218, 5.29713791014720389221627628941, 6.16840818826871691353069623739, 6.75676800327108810247954825626, 7.75661284561345727590061338149, 8.593871174727172033286930528651, 9.146507633973911148914873624485