L(s) = 1 | − 3-s + 1.34·5-s + 2.41·7-s + 9-s − 5.94·11-s + 3.69·13-s − 1.34·15-s − 0.162·17-s − 2.41·21-s − 6.57·23-s − 3.18·25-s − 27-s − 2.38·29-s + 2.59·31-s + 5.94·33-s + 3.24·35-s + 6.94·37-s − 3.69·39-s + 0.361·41-s − 11.4·43-s + 1.34·45-s − 6.87·47-s − 1.18·49-s + 0.162·51-s − 4.02·53-s − 8.00·55-s − 12.0·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.602·5-s + 0.911·7-s + 0.333·9-s − 1.79·11-s + 1.02·13-s − 0.347·15-s − 0.0394·17-s − 0.526·21-s − 1.37·23-s − 0.636·25-s − 0.192·27-s − 0.443·29-s + 0.466·31-s + 1.03·33-s + 0.549·35-s + 1.14·37-s − 0.591·39-s + 0.0565·41-s − 1.74·43-s + 0.200·45-s − 1.00·47-s − 0.169·49-s + 0.0227·51-s − 0.552·53-s − 1.07·55-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 5.94T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 + 0.162T + 17T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 0.361T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 6.18T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 9.17T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 - 2.02T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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
−8.069375364695725689479553345222, −7.42375116889764974047233680407, −6.24197816836159136822392556541, −5.86770366109587897505885471216, −5.07479395163237082930837941238, −4.48601034325866799838275170460, −3.36253480616856678528579278897, −2.23603256558544407254157460930, −1.49634178269012634700314347759, 0,
1.49634178269012634700314347759, 2.23603256558544407254157460930, 3.36253480616856678528579278897, 4.48601034325866799838275170460, 5.07479395163237082930837941238, 5.86770366109587897505885471216, 6.24197816836159136822392556541, 7.42375116889764974047233680407, 8.069375364695725689479553345222