L(s) = 1 | + (−0.453 + 0.891i)2-s + (−0.348 − 2.20i)3-s + (−0.587 − 0.809i)4-s + (−1.49 − 1.66i)5-s + (2.11 + 0.688i)6-s + (−3.91 − 0.620i)7-s + (0.987 − 0.156i)8-s + (−1.87 + 0.608i)9-s + (2.15 − 0.579i)10-s + (2.83 − 1.71i)11-s + (−1.57 + 1.57i)12-s + (2.65 + 1.35i)13-s + (2.33 − 3.20i)14-s + (−3.13 + 3.87i)15-s + (−0.309 + 0.951i)16-s + (4.15 − 2.11i)17-s + ⋯ |
L(s) = 1 | + (−0.321 + 0.630i)2-s + (−0.201 − 1.27i)3-s + (−0.293 − 0.404i)4-s + (−0.669 − 0.742i)5-s + (0.865 + 0.281i)6-s + (−1.48 − 0.234i)7-s + (0.349 − 0.0553i)8-s + (−0.623 + 0.202i)9-s + (0.682 − 0.183i)10-s + (0.856 − 0.516i)11-s + (−0.454 + 0.454i)12-s + (0.736 + 0.375i)13-s + (0.623 − 0.857i)14-s + (−0.809 + 1.00i)15-s + (−0.0772 + 0.237i)16-s + (1.00 − 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0401 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0401 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.438447 - 0.456418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.438447 - 0.456418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 5 | \( 1 + (1.49 + 1.66i)T \) |
| 11 | \( 1 + (-2.83 + 1.71i)T \) |
good | 3 | \( 1 + (0.348 + 2.20i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (3.91 + 0.620i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 1.35i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.15 + 2.11i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 1.88i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.14 + 5.14i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.0660 + 0.0479i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.600 + 1.84i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.916 - 5.78i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.64 + 2.25i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (2.07 - 2.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-12.1 + 1.92i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.67 + 3.28i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.48 + 2.04i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 0.554i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.34 + 2.34i)T - 67iT^{2} \) |
| 71 | \( 1 + (1.98 - 6.11i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.22 - 7.72i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (3.80 + 11.7i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.86 - 13.4i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 5.99iT - 89T^{2} \) |
| 97 | \( 1 + (-9.78 - 4.98i)T + (57.0 + 78.4i)T^{2} \) |
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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
−13.38116113651958062785527787452, −12.42462457710530849732084929465, −11.72742684660488180390314220470, −9.947724125940282963976837362633, −8.825816184149674244120754609375, −7.74751181303745681092257962240, −6.72201421982105016520243619751, −5.90063971623539056464159758999, −3.79429324514941577669670001810, −0.870273879123255464743686064895,
3.32129729147866175013606167277, 3.94882029045176690936848951837, 5.91203764249353678005366474511, 7.40602729479999098079014574138, 9.069235489060812641761891644953, 9.887367658205328761643750501067, 10.56126072114446486970983712085, 11.67320654198657579939330479651, 12.56274274496940389998164418657, 13.97935529975814202752841880797