L(s) = 1 | + (1 − 1.73i)4-s + (2.5 + 0.866i)7-s + (3.5 + 6.06i)13-s + (−1.99 − 3.46i)16-s + (3.5 − 6.06i)19-s − 5·25-s + (4 − 3.46i)28-s + (3.5 − 6.06i)31-s + (0.5 − 0.866i)37-s + (−2.5 + 4.33i)43-s + (5.5 + 4.33i)49-s + 14·52-s + (−7 − 12.1i)61-s − 7.99·64-s + (−5.5 + 9.52i)67-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (0.944 + 0.327i)7-s + (0.970 + 1.68i)13-s + (−0.499 − 0.866i)16-s + (0.802 − 1.39i)19-s − 25-s + (0.755 − 0.654i)28-s + (0.628 − 1.08i)31-s + (0.0821 − 0.142i)37-s + (−0.381 + 0.660i)43-s + (0.785 + 0.618i)49-s + 1.94·52-s + (−0.896 − 1.55i)61-s − 0.999·64-s + (−0.671 + 1.16i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80159 - 0.407253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80159 - 0.407253i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.00618687603374488100522142787, −9.681627654740260582367234391236, −9.118792271025037367087670882804, −8.017022069291099811284592791469, −6.94145440879291088465465266815, −6.14595830660429113042814811759, −5.14099481787855597373490971154, −4.19359300362076988884737933878, −2.42952941470636248367231201601, −1.35894369580060168673200234568,
1.50383301377128624658897602765, 3.08517246852557650342965615941, 3.93017140693550642543459058897, 5.29940359658242321547915691426, 6.24926640933444087429668999049, 7.60965164553810521912058029929, 7.938533874516159532729217850229, 8.754741901892339896806207403149, 10.24505384462205628742717488056, 10.75743606721848065063443148425