L(s) = 1 | + (1 − 1.73i)4-s + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s + (−1.99 − 3.46i)16-s + 6·17-s − 19-s + (0.999 + 1.73i)20-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s − 3.99·28-s + (−4.5 − 7.79i)29-s + (0.5 − 0.866i)31-s + 1.99·35-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s + (−0.499 − 0.866i)16-s + 1.45·17-s − 0.229·19-s + (0.223 + 0.387i)20-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.755·28-s + (−0.835 − 1.44i)29-s + (0.0898 − 0.155i)31-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02546 - 0.860470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02546 - 0.860470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (-2 - 3.46i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98682490534469542709071721179, −10.20257874989331194841245820005, −9.591002042220460826834510173618, −8.001586516442057116452193189702, −7.42259803845078450566992005547, −6.02836041327421779299848977799, −5.65446331133247631398115910295, −3.91093827247112766791027217507, −2.78601023189117576833444901547, −0.902664552866366839546307342608,
2.03719337583772445132671745598, 3.33271500820978335412958428860, 4.45589045072803723784453948044, 5.80033750515266137045392067283, 6.86883299896520198180623311240, 7.78568872840796869157171119700, 8.638665127483770727209962501065, 9.510039272189553421049646817551, 10.65747478322144581995655218109, 11.64565840240670253074001782993