L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 4.47·7-s + 3·8-s + 9-s + 3.23·11-s + 12-s − 3.38·13-s + 4.47·14-s − 16-s + 2.85·17-s − 18-s − 3.23·19-s + 4.47·21-s − 3.23·22-s + 4.47·23-s − 3·24-s + 3.38·26-s − 27-s + 4.47·28-s + 4.38·29-s + 7.23·31-s − 5·32-s − 3.23·33-s − 2.85·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.5·4-s + 0.408·6-s − 1.69·7-s + 1.06·8-s + 0.333·9-s + 0.975·11-s + 0.288·12-s − 0.937·13-s + 1.19·14-s − 0.250·16-s + 0.692·17-s − 0.235·18-s − 0.742·19-s + 0.975·21-s − 0.689·22-s + 0.932·23-s − 0.612·24-s + 0.663·26-s − 0.192·27-s + 0.845·28-s + 0.813·29-s + 1.29·31-s − 0.883·32-s − 0.563·33-s − 0.489·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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
−9.139631341288759573391165811585, −8.189373589478111696146722531933, −7.08906283646487937755354944775, −6.67003064948681231442910331723, −5.72969960009539445483473881836, −4.71070563011167120070767177093, −3.87964560450793591322330078854, −2.80688919068116824546794244027, −1.13868330344096009920645055642, 0,
1.13868330344096009920645055642, 2.80688919068116824546794244027, 3.87964560450793591322330078854, 4.71070563011167120070767177093, 5.72969960009539445483473881836, 6.67003064948681231442910331723, 7.08906283646487937755354944775, 8.189373589478111696146722531933, 9.139631341288759573391165811585