L(s) = 1 | + (−0.104 + 0.391i)2-s + (0.735 + 1.56i)3-s + (1.58 + 0.917i)4-s + (0.537 − 2.00i)5-s + (−0.690 + 0.123i)6-s + (−1.39 − 1.39i)7-s + (−1.09 + 1.09i)8-s + (−1.91 + 2.30i)9-s + (0.728 + 0.420i)10-s + (−2.68 − 0.720i)11-s + (−0.270 + 3.16i)12-s + (3.47 + 0.957i)13-s + (0.692 − 0.399i)14-s + (3.54 − 0.632i)15-s + (1.52 + 2.63i)16-s + (−2.88 − 5.00i)17-s + ⋯ |
L(s) = 1 | + (−0.0741 + 0.276i)2-s + (0.424 + 0.905i)3-s + (0.794 + 0.458i)4-s + (0.240 − 0.897i)5-s + (−0.282 + 0.0503i)6-s + (−0.527 − 0.527i)7-s + (−0.388 + 0.388i)8-s + (−0.639 + 0.768i)9-s + (0.230 + 0.133i)10-s + (−0.810 − 0.217i)11-s + (−0.0781 + 0.914i)12-s + (0.964 + 0.265i)13-s + (0.184 − 0.106i)14-s + (0.914 − 0.163i)15-s + (0.380 + 0.658i)16-s + (−0.700 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12036 + 0.554003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12036 + 0.554003i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.735 - 1.56i)T \) |
| 13 | \( 1 + (-3.47 - 0.957i)T \) |
good | 2 | \( 1 + (0.104 - 0.391i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.537 + 2.00i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.39 + 1.39i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.68 + 0.720i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.88 + 5.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.00 + 1.07i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 + (0.0236 - 0.0136i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.49 + 1.20i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-6.63 + 1.77i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.64 - 6.64i)T + 41iT^{2} \) |
| 43 | \( 1 + 6.27iT - 43T^{2} \) |
| 47 | \( 1 + (1.14 + 4.26i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 - 8.65iT - 53T^{2} \) |
| 59 | \( 1 + (-3.56 - 13.2i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 9.28T + 61T^{2} \) |
| 67 | \( 1 + (4.68 - 4.68i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.46 - 9.20i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.696 - 0.696i)T + 73iT^{2} \) |
| 79 | \( 1 + (6.46 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.21 + 0.860i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.569 - 2.12i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-8.64 + 8.64i)T - 97iT^{2} \) |
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
−13.52796174747060265484302805740, −12.93761273464113870854665509660, −11.36328590708868133970455129311, −10.62125915061154681465622667475, −9.225360770364121439488280383795, −8.498690935312156502569185042927, −7.20048984497195776981508238837, −5.72615184909550819072680194528, −4.28806953113095585957754948384, −2.76341638020609746313259557819,
2.07567010018075929808332385626, 3.16328374863487413581451707055, 6.05673780334691000406803184888, 6.50658094405503234418677971907, 7.82795984111081795469267199252, 9.128336467235363134929496126871, 10.54370112925204189045102251989, 11.11862459998043485600114418798, 12.57652545324340862915523905621, 13.11384563351456860009877246994