L(s) = 1 | + (0.489 + 0.872i)5-s + (−0.822 + 0.569i)7-s + (0.169 + 0.985i)11-s + (0.700 − 0.713i)13-s + (−0.929 + 0.369i)17-s + (0.206 − 0.978i)19-s + (−0.316 − 0.948i)23-s + (−0.521 + 0.853i)25-s + (−0.316 + 0.948i)29-s + (−0.421 − 0.906i)31-s + (−0.898 − 0.438i)35-s + (0.243 − 0.969i)37-s + (0.644 − 0.764i)41-s + (−0.455 − 0.890i)43-s + (−0.0944 + 0.995i)47-s + ⋯ |
L(s) = 1 | + (0.489 + 0.872i)5-s + (−0.822 + 0.569i)7-s + (0.169 + 0.985i)11-s + (0.700 − 0.713i)13-s + (−0.929 + 0.369i)17-s + (0.206 − 0.978i)19-s + (−0.316 − 0.948i)23-s + (−0.521 + 0.853i)25-s + (−0.316 + 0.948i)29-s + (−0.421 − 0.906i)31-s + (−0.898 − 0.438i)35-s + (0.243 − 0.969i)37-s + (0.644 − 0.764i)41-s + (−0.455 − 0.890i)43-s + (−0.0944 + 0.995i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5604502758 - 0.4801819210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5604502758 - 0.4801819210i\) |
\(L(1)\) |
\(\approx\) |
\(0.9069607491 + 0.1158460429i\) |
\(L(1)\) |
\(\approx\) |
\(0.9069607491 + 0.1158460429i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 \) |
good | 5 | \( 1 + (0.489 + 0.872i)T \) |
| 7 | \( 1 + (-0.822 + 0.569i)T \) |
| 11 | \( 1 + (0.169 + 0.985i)T \) |
| 13 | \( 1 + (0.700 - 0.713i)T \) |
| 17 | \( 1 + (-0.929 + 0.369i)T \) |
| 19 | \( 1 + (0.206 - 0.978i)T \) |
| 23 | \( 1 + (-0.316 - 0.948i)T \) |
| 29 | \( 1 + (-0.316 + 0.948i)T \) |
| 31 | \( 1 + (-0.421 - 0.906i)T \) |
| 37 | \( 1 + (0.243 - 0.969i)T \) |
| 41 | \( 1 + (0.644 - 0.764i)T \) |
| 43 | \( 1 + (-0.455 - 0.890i)T \) |
| 47 | \( 1 + (-0.0944 + 0.995i)T \) |
| 53 | \( 1 + (-0.752 + 0.658i)T \) |
| 59 | \( 1 + (-0.929 - 0.369i)T \) |
| 61 | \( 1 + (-0.672 - 0.739i)T \) |
| 67 | \( 1 + (0.489 - 0.872i)T \) |
| 71 | \( 1 + (-0.843 - 0.537i)T \) |
| 73 | \( 1 + (0.997 + 0.0756i)T \) |
| 79 | \( 1 + (-0.974 + 0.225i)T \) |
| 83 | \( 1 + (0.0189 - 0.999i)T \) |
| 89 | \( 1 + (-0.988 + 0.150i)T \) |
| 97 | \( 1 + (0.421 - 0.906i)T \) |
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
−18.60728528181976589046132604425, −17.9622356517978889637309735366, −17.09168768530447330375376357911, −16.497270752905484566145039724277, −16.16931873941935472009059293922, −15.467417084734558318169095525412, −14.27714774389891890321074260654, −13.67276835756977973113009071804, −13.338841271842122833157028167840, −12.63487467804470939504347448155, −11.65923470672150152005686269241, −11.21999844511831706735242169274, −10.151620815993122217736129273886, −9.63310472880556585224483372852, −8.94083020032760065583741070518, −8.33306430757912915729450936103, −7.47285543038490142646233508674, −6.38935828855425724334916077692, −6.1354283831342920467350372074, −5.22549991440192538807411435732, −4.27085086719541831494419105385, −3.70177833843668531442856047596, −2.836938957755860487773263252328, −1.65440283899165742884069402180, −1.06604343387372346181050606644,
0.20877811055626074574137818575, 1.748751711165495823815212674742, 2.40376615596730388698076902837, 3.10128557942411445035763402487, 3.916263290027504799179265722745, 4.85240857961954820166310043816, 5.84413657779695603928171488050, 6.30556652290848761277775639276, 7.00772672201350079133452246138, 7.67495858689823721460401950992, 8.85085220481249210776390891332, 9.273182180562347731163821273080, 10.02827437629892255402250498850, 10.82438155053234179561612658084, 11.17757209097773892441355464059, 12.45146934462947776569146051284, 12.73878321174310628554936429346, 13.526580413009669047382422677332, 14.25853113653634204143385919240, 15.10558830360022019772691292907, 15.45468746625549569109915963107, 16.13052360380762343416659848331, 17.18438221578641099186945910404, 17.6950663331941738691671679995, 18.403690824859484109067902355052