| 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 | \( 1 + (-9.78 + 4.98i)T + (57.0 - 78.4i)T^{2} \) |
| 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
−13.97935529975814202752841880797, −12.56274274496940389998164418657, −11.67320654198657579939330479651, −10.56126072114446486970983712085, −9.887367658205328761643750501067, −9.069235489060812641761891644953, −7.40602729479999098079014574138, −5.91203764249353678005366474511, −3.94882029045176690936848951837, −3.32129729147866175013606167277,
0.870273879123255464743686064895, 3.79429324514941577669670001810, 5.90063971623539056464159758999, 6.72201421982105016520243619751, 7.74751181303745681092257962240, 8.825816184149674244120754609375, 9.947724125940282963976837362633, 11.72742684660488180390314220470, 12.42462457710530849732084929465, 13.38116113651958062785527787452
