| L(s) = 1 | − 2-s + 4-s − 3.37·7-s − 8-s + 11-s − 2·13-s + 3.37·14-s + 16-s + 1.37·17-s + 0.627·19-s − 22-s + 2.74·23-s + 2·26-s − 3.37·28-s − 1.37·29-s + 3.37·31-s − 32-s − 1.37·34-s − 9.37·37-s − 0.627·38-s + 11.4·41-s + 4·43-s + 44-s − 2.74·46-s + 2.74·47-s + 4.37·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.27·7-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.901·14-s + 0.250·16-s + 0.332·17-s + 0.144·19-s − 0.213·22-s + 0.572·23-s + 0.392·26-s − 0.637·28-s − 0.254·29-s + 0.605·31-s − 0.176·32-s − 0.235·34-s − 1.54·37-s − 0.101·38-s + 1.79·41-s + 0.609·43-s + 0.150·44-s − 0.404·46-s + 0.400·47-s + 0.624·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
| 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
−7.79294763549993325217619808678, −7.26103110883045301954338705360, −6.53832530644259488053929343841, −5.95602454770431659440922950217, −5.05554384694337290805841813534, −3.97151016297051327878671560357, −3.14533124261814314153039276429, −2.43455653025072204342386265506, −1.16253671815266753101452741456, 0,
1.16253671815266753101452741456, 2.43455653025072204342386265506, 3.14533124261814314153039276429, 3.97151016297051327878671560357, 5.05554384694337290805841813534, 5.95602454770431659440922950217, 6.53832530644259488053929343841, 7.26103110883045301954338705360, 7.79294763549993325217619808678
