L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s − 1.38·5-s + 1.61·6-s + 2.23·8-s + 9-s + 2.23·10-s + 11-s − 0.618·12-s + 1.61·13-s + 1.38·15-s − 4.85·16-s − 3.47·17-s − 1.61·18-s + 0.236·19-s − 0.854·20-s − 1.61·22-s + 23-s − 2.23·24-s − 3.09·25-s − 2.61·26-s − 27-s + 6.23·29-s − 2.23·30-s − 4.70·31-s + 3.38·32-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.618·5-s + 0.660·6-s + 0.790·8-s + 0.333·9-s + 0.707·10-s + 0.301·11-s − 0.178·12-s + 0.448·13-s + 0.356·15-s − 1.21·16-s − 0.842·17-s − 0.381·18-s + 0.0541·19-s − 0.190·20-s − 0.344·22-s + 0.208·23-s − 0.456·24-s − 0.618·25-s − 0.513·26-s − 0.192·27-s + 1.15·29-s − 0.408·30-s − 0.845·31-s + 0.597·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 - 0.236T + 19T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 3.09T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 + 6.23T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 - 0.0557T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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
−8.501954898033375761711308173917, −7.51543241308337602333079311559, −7.00939397559259908016658271317, −6.20104465273456098967994770912, −5.16756878803702128324797287319, −4.35600052022758166203156529886, −3.62944497746666368708655234986, −2.15906198549891827401582384598, −1.06787939464176201037385868809, 0,
1.06787939464176201037385868809, 2.15906198549891827401582384598, 3.62944497746666368708655234986, 4.35600052022758166203156529886, 5.16756878803702128324797287319, 6.20104465273456098967994770912, 7.00939397559259908016658271317, 7.51543241308337602333079311559, 8.501954898033375761711308173917