L(s) = 1 | − 3-s − 0.614·5-s + 3.46·7-s + 9-s − 3.80·11-s − 2.93·13-s + 0.614·15-s + 3.15·17-s + 5.58·19-s − 3.46·21-s + 0.574·23-s − 4.62·25-s − 27-s − 3.25·29-s + 1.74·31-s + 3.80·33-s − 2.12·35-s − 4.35·37-s + 2.93·39-s − 1.51·41-s − 9.97·43-s − 0.614·45-s + 8.29·47-s + 4.99·49-s − 3.15·51-s + 0.465·53-s + 2.34·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.275·5-s + 1.30·7-s + 0.333·9-s − 1.14·11-s − 0.812·13-s + 0.158·15-s + 0.765·17-s + 1.28·19-s − 0.755·21-s + 0.119·23-s − 0.924·25-s − 0.192·27-s − 0.603·29-s + 0.313·31-s + 0.663·33-s − 0.359·35-s − 0.716·37-s + 0.469·39-s − 0.236·41-s − 1.52·43-s − 0.0916·45-s + 1.20·47-s + 0.713·49-s − 0.441·51-s + 0.0638·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 0.614T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 - 0.574T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 - 0.465T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 8.39T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57500697412935966083318325002, −7.03159731142673937101313929924, −5.84766499767628997238390920654, −5.22192149346962619483050674292, −4.98489599754750211636755983909, −4.02933269567062350680807549684, −3.09441225749019886403543995806, −2.12362682509620663223155997689, −1.22707057726017983380740648444, 0,
1.22707057726017983380740648444, 2.12362682509620663223155997689, 3.09441225749019886403543995806, 4.02933269567062350680807549684, 4.98489599754750211636755983909, 5.22192149346962619483050674292, 5.84766499767628997238390920654, 7.03159731142673937101313929924, 7.57500697412935966083318325002