L(s) = 1 | + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (0.996 − 0.0871i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.0871 + 0.996i)29-s + (−0.766 − 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (−0.766 + 0.642i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (0.996 − 0.0871i)13-s + (0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.0871 + 0.996i)29-s + (−0.766 − 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.422 − 0.906i)43-s + (−0.766 + 0.642i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6617178724 + 0.5617178859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6617178724 + 0.5617178859i\) |
\(L(1)\) |
\(\approx\) |
\(0.9129012155 - 0.06912518063i\) |
\(L(1)\) |
\(\approx\) |
\(0.9129012155 - 0.06912518063i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.0871 - 0.996i)T \) |
| 11 | \( 1 + (-0.996 - 0.0871i)T \) |
| 13 | \( 1 + (0.996 - 0.0871i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.258 + 0.965i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.0871 + 0.996i)T \) |
| 31 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.642 - 0.766i)T \) |
| 43 | \( 1 + (-0.422 - 0.906i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (0.965 - 0.258i)T \) |
| 59 | \( 1 + (-0.819 + 0.573i)T \) |
| 61 | \( 1 + (-0.0871 + 0.996i)T \) |
| 67 | \( 1 + (-0.422 + 0.906i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.0871 + 0.996i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.92188813031832564809031058741, −16.88034493275412470478682716569, −16.05684638441553794893275969713, −15.57764617706136060733027042834, −15.09431903701861416756655833162, −14.19281983867658106868104796926, −13.59035269041165011880645096139, −13.22792451510646999385096676526, −12.17698291827473740916404371588, −11.419334836086641505270143839615, −11.06908197106475492694946431774, −10.28148823020032144570802122909, −9.6795643321005365646953637390, −8.96571222830784939491853285368, −7.87952584983478870269044890602, −7.65227425417090310421309727845, −6.73338506192979894698275642460, −6.16706060522249052850979550297, −5.31314413750779219562560390355, −4.68803486457350014609078960623, −3.52479629049486438457717269540, −3.1760176885109729755816759334, −2.35039506520995254445737930618, −1.48578240583872113321840244804, −0.23756008843350668434030165015,
0.95513887985740601048210483702, 1.66960033263249029988077345523, 2.523405944708493509384835008412, 3.72872063672948720366360512374, 3.93860192771959034666889698566, 5.097961455338736761971676247572, 5.59330860875158438325601672460, 6.1400652566044722821404347783, 7.262878581599517331147914900341, 7.93652983110130015634128840722, 8.52931624804975719810813393925, 8.97785679927406108182139871071, 10.01382959507543696015181437483, 10.51117657959507423059366193632, 11.18191653671741731238827739293, 12.152494641728200398234907732352, 12.61928927405668612111099787028, 13.106984971727067834977301893522, 13.86962530208586671430202360051, 14.549024419874956988235470430808, 15.36738776748736533498094183436, 16.05459276989923249417320885001, 16.44135168928771271919842201609, 17.03551030608620807063892693921, 17.95711294091120543436241613799