L(s) = 1 | + (0.681 − 1.23i)2-s + (−1.07 − 1.68i)4-s + 5-s − 1.41i·7-s + (−2.82 + 0.175i)8-s + (0.681 − 1.23i)10-s − 6.37i·11-s + 3.54i·13-s + (−1.75 − 0.964i)14-s + (−1.70 + 3.61i)16-s − 3.92i·17-s + 1.27·19-s + (−1.07 − 1.68i)20-s + (−7.89 − 4.34i)22-s + 6.28·23-s + ⋯ |
L(s) = 1 | + (0.482 − 0.876i)2-s + (−0.535 − 0.844i)4-s + 0.447·5-s − 0.534i·7-s + (−0.998 + 0.0619i)8-s + (0.215 − 0.391i)10-s − 1.92i·11-s + 0.982i·13-s + (−0.468 − 0.257i)14-s + (−0.426 + 0.904i)16-s − 0.952i·17-s + 0.292·19-s + (−0.239 − 0.377i)20-s + (−1.68 − 0.925i)22-s + 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.793827 - 1.42372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.793827 - 1.42372i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.681 + 1.23i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 6.37iT - 11T^{2} \) |
| 13 | \( 1 - 3.54iT - 13T^{2} \) |
| 17 | \( 1 + 3.92iT - 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + 9.00T + 29T^{2} \) |
| 31 | \( 1 - 3.92iT - 31T^{2} \) |
| 37 | \( 1 - 2.51iT - 37T^{2} \) |
| 41 | \( 1 - 5.27iT - 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 + 0.313iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 + 0.990T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 8.18iT - 79T^{2} \) |
| 83 | \( 1 - 5.02iT - 83T^{2} \) |
| 89 | \( 1 + 0.386iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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
−11.19032506031163010866493225843, −10.50319971195681397074374701354, −9.295342005896128010923598614755, −8.805668902747117254702106405792, −7.17785677418869663149500412538, −6.01772574654796768514878768774, −5.11713347030823826016667275002, −3.82187126917501070753920460209, −2.77504146160861910920654477091, −1.04825444583806461937419346741,
2.32439493219263104552664642614, 3.87592326277221873390286920757, 5.10978474655816132414590688472, 5.81391628355597829055086288547, 7.04666182253185737815682282431, 7.72605559165526206006486177472, 8.957164498341979396582168697380, 9.656311119849090769827092636990, 10.79352902996420200637114140331, 12.21073748633276887933479350986