L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s + 23-s + 27-s + 29-s − 31-s + (0.5 − 0.866i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s + 23-s + 27-s + 29-s − 31-s + (0.5 − 0.866i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1282030053 - 0.3242722177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1282030053 - 0.3242722177i\) |
\(L(1)\) |
\(\approx\) |
\(0.7346486735 + 0.01757695584i\) |
\(L(1)\) |
\(\approx\) |
\(0.7346486735 + 0.01757695584i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.25391530376411015779674509332, −20.0404942442650434598316671659, −19.23582183863483917254106365902, −18.60651223757661291078833194514, −18.05783923113335503619570924795, −17.342931188654757398126691363787, −16.56969231253844441639253084111, −15.619070405707505658682073888353, −15.000541157898644713544735180608, −13.96844612201063210054842290649, −13.20920364895054567527245282581, −12.628824102681577987653153571002, −11.79888506851540914580893460081, −11.048057880070086614973327325543, −10.5561325264288962306001876472, −9.05977204080199848658596845710, −8.54681323775306989852942664920, −7.71683601190036219151086388590, −6.75798148149448425182286628813, −6.12996384566214724502757023328, −5.18349622626965211043344731594, −4.58082009679044742344268058218, −3.058097928089451833554544487765, −2.15506270052824285493091463263, −1.46582054668694221369086622194,
0.140316433629272052398773679503, 1.366981714856338164184892240936, 2.86948217001321470292461675038, 3.595685689210069721839675340772, 4.69550024755669309105934792375, 5.0986906203542467341830060627, 6.10415538029871116126400087637, 7.049668782276868368286666037106, 8.02098048489408556989874754080, 8.716215284709488448408960460677, 9.87540804334821337716155960084, 10.429844512424568753893544681708, 11.01987859242272327174228110674, 11.72025645590125055399679017959, 12.88131834358562110032211171574, 13.4545851321709184587006501667, 14.55602408619804606217495660872, 15.079337780184147661027867974, 16.05992573748339494779118751843, 16.42932500460763241477496352034, 17.44593515345570167705594137716, 17.91262775120202183342675685748, 18.69995089424796506210748108940, 19.974135866895479212362864982145, 20.51710894809338461083743026501