L(s) = 1 | − 2-s + 4-s − 2.71·5-s + 1.67·7-s − 8-s + 2.71·10-s + 4.37·11-s − 2.34·13-s − 1.67·14-s + 16-s + 4.09·17-s − 7.26·19-s − 2.71·20-s − 4.37·22-s − 0.0981·23-s + 2.34·25-s + 2.34·26-s + 1.67·28-s + 5.89·29-s − 1.88·31-s − 32-s − 4.09·34-s − 4.54·35-s − 3.13·37-s + 7.26·38-s + 2.71·40-s + 0.419·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.21·5-s + 0.634·7-s − 0.353·8-s + 0.857·10-s + 1.31·11-s − 0.651·13-s − 0.448·14-s + 0.250·16-s + 0.993·17-s − 1.66·19-s − 0.606·20-s − 0.932·22-s − 0.0204·23-s + 0.469·25-s + 0.460·26-s + 0.317·28-s + 1.09·29-s − 0.337·31-s − 0.176·32-s − 0.702·34-s − 0.768·35-s − 0.515·37-s + 1.17·38-s + 0.428·40-s + 0.0655·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 23 | \( 1 + 0.0981T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 - 0.419T + 41T^{2} \) |
| 43 | \( 1 + 0.0900T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.63T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 - 1.15T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 + 4.78T + 97T^{2} \) |
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
−7.70467613584467504964946055678, −6.85060196271813872129122785284, −6.45257242179096126207967959475, −5.37892858471176296061806771162, −4.47844527762173951485565500078, −3.95296803494980967245717407100, −3.11763154553051407841753694511, −2.02447755525788029964145097941, −1.13089887684212813256058492753, 0,
1.13089887684212813256058492753, 2.02447755525788029964145097941, 3.11763154553051407841753694511, 3.95296803494980967245717407100, 4.47844527762173951485565500078, 5.37892858471176296061806771162, 6.45257242179096126207967959475, 6.85060196271813872129122785284, 7.70467613584467504964946055678