L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7246886684\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7246886684\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (0.987 - 0.156i)T \) |
good | 7 | \( 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.69 + 0.550i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.97iT - T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−8.482574747722734675760319290621, −7.82117781790747488862686464160, −7.05719413122585773083628233118, −6.36662764978008248958202451359, −5.58948555045842812317350395169, −4.96964983808520353156904825269, −4.46412963115344089580883952733, −2.80091412680833316876344820724, −2.12352777116065178785801289378, −0.43695981798257153839927964775,
1.43214921403448145097296640256, 2.42499591466296938701961123239, 2.90754120762772124436827162102, 4.04989958509836348092191642003, 4.94217013518638586312485494492, 5.56082314087060207625491007027, 6.56182604501615659225934368215, 7.62069204225928416710377310953, 7.83356033690376033295804298731, 9.128175038588413475063138372960