L(s) = 1 | + (−1.23 + 2.13i)2-s + (−1.66 − 0.486i)3-s + (−2.02 − 3.51i)4-s + (−1.82 − 3.16i)5-s + (3.08 − 2.94i)6-s + 5.05·8-s + (2.52 + 1.61i)9-s + 9.00·10-s + (−0.203 + 0.351i)11-s + (1.66 + 6.82i)12-s + (−0.243 − 0.421i)13-s + (1.5 + 6.15i)15-s + (−2.16 + 3.74i)16-s − 4.85·17-s + (−6.55 + 3.39i)18-s + 1.97·19-s + ⋯ |
L(s) = 1 | + (−0.869 + 1.50i)2-s + (−0.959 − 0.280i)3-s + (−1.01 − 1.75i)4-s + (−0.817 − 1.41i)5-s + (1.25 − 1.20i)6-s + 1.78·8-s + (0.842 + 0.538i)9-s + 2.84·10-s + (−0.0612 + 0.106i)11-s + (0.479 + 1.96i)12-s + (−0.0675 − 0.116i)13-s + (0.387 + 1.58i)15-s + (−0.540 + 0.936i)16-s − 1.17·17-s + (−1.54 + 0.800i)18-s + 0.452·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0184495 + 0.166964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0184495 + 0.166964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.66 + 0.486i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.82 + 3.16i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.203 - 0.351i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.243 + 0.421i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 1.97T + 19T^{2} \) |
| 23 | \( 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.82 - 6.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.51 - 6.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.16 + 2.01i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.15 - 5.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 + (-3.05 - 5.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.01 - 6.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.80 + 3.11i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 + (4.08 - 7.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.08 + 10.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + (4.74 - 8.21i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.50446706200717068462634706446, −10.46490894885566630577453123168, −9.352396705448427796019144344093, −8.602571486765392860444008372943, −7.83576806232338435098099932310, −7.02097442031495775565101248420, −6.07844491651382466802501592917, −5.04780195225016080744773969046, −4.47019476306813399485602280640, −1.12318752379338300854307955239,
0.19139264242936706112214232654, 2.23928421073305948393797107801, 3.52449501473584548064207892245, 4.29083404096157381877835209776, 6.07655946077484552628899555341, 7.19692660394761104574555960288, 8.050339845206914095056591453087, 9.443881285676105181564109875903, 10.03494745051418794191444737423, 10.97145414777760852905209502547