L(s) = 1 | + (−0.919 − 0.391i)2-s + (0.632 + 0.774i)3-s + (0.692 + 0.721i)4-s + (−0.0402 − 0.999i)5-s + (−0.278 − 0.960i)6-s + (−0.987 + 0.160i)7-s + (−0.354 − 0.935i)8-s + (−0.200 + 0.979i)9-s + (−0.354 + 0.935i)10-s + (−0.845 − 0.534i)11-s + (−0.120 + 0.992i)12-s + (0.278 − 0.960i)13-s + (0.970 + 0.239i)14-s + (0.748 − 0.663i)15-s + (−0.0402 + 0.999i)16-s + (0.970 − 0.239i)17-s + ⋯ |
L(s) = 1 | + (−0.919 − 0.391i)2-s + (0.632 + 0.774i)3-s + (0.692 + 0.721i)4-s + (−0.0402 − 0.999i)5-s + (−0.278 − 0.960i)6-s + (−0.987 + 0.160i)7-s + (−0.354 − 0.935i)8-s + (−0.200 + 0.979i)9-s + (−0.354 + 0.935i)10-s + (−0.845 − 0.534i)11-s + (−0.120 + 0.992i)12-s + (0.278 − 0.960i)13-s + (0.970 + 0.239i)14-s + (0.748 − 0.663i)15-s + (−0.0402 + 0.999i)16-s + (0.970 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2240452918 - 0.5018981870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2240452918 - 0.5018981870i\) |
\(L(1)\) |
\(\approx\) |
\(0.6223988943 - 0.1524084552i\) |
\(L(1)\) |
\(\approx\) |
\(0.6223988943 - 0.1524084552i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (-0.919 - 0.391i)T \) |
| 3 | \( 1 + (0.632 + 0.774i)T \) |
| 5 | \( 1 + (-0.0402 - 0.999i)T \) |
| 7 | \( 1 + (-0.987 + 0.160i)T \) |
| 11 | \( 1 + (-0.845 - 0.534i)T \) |
| 13 | \( 1 + (0.278 - 0.960i)T \) |
| 17 | \( 1 + (0.970 - 0.239i)T \) |
| 19 | \( 1 + (-0.996 - 0.0804i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.948 - 0.316i)T \) |
| 31 | \( 1 + (0.799 - 0.600i)T \) |
| 37 | \( 1 + (-0.428 - 0.903i)T \) |
| 41 | \( 1 + (-0.885 - 0.464i)T \) |
| 43 | \( 1 + (0.845 - 0.534i)T \) |
| 47 | \( 1 + (-0.428 + 0.903i)T \) |
| 53 | \( 1 + (0.632 - 0.774i)T \) |
| 59 | \( 1 + (-0.692 + 0.721i)T \) |
| 61 | \( 1 + (-0.568 + 0.822i)T \) |
| 67 | \( 1 + (0.120 - 0.992i)T \) |
| 71 | \( 1 + (0.354 + 0.935i)T \) |
| 73 | \( 1 + (0.278 + 0.960i)T \) |
| 83 | \( 1 + (0.692 + 0.721i)T \) |
| 89 | \( 1 + (-0.354 + 0.935i)T \) |
| 97 | \( 1 + (0.568 - 0.822i)T \) |
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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
−31.18782024429143671079914228262, −29.883353165287253943266547789757, −29.28198879489977474580161765070, −28.044725167764028870268503109623, −26.45504360026874976150528345574, −25.94151815566062482276877896851, −25.3448259881013093221337622640, −23.714699323419554211465621503951, −23.15741316662524489685270965373, −21.22076786363975658508434255147, −19.84637028096088940795184911994, −18.94139232312938578295320439849, −18.44304518503837746812837119525, −17.11802855626229649527025834648, −15.68214508531592239707832190602, −14.698556999575344789765941961573, −13.53582748344248632676057193543, −12.00166554872353840431101027486, −10.44066925961418162489474220214, −9.45529809321928318438308619617, −7.9882752593510202148866877069, −7.0013599274810898867803074459, −6.15495865702598995945369761152, −3.252773934254870768015161285015, −1.89748842145217392828857748873,
0.30898546586682104464500625590, 2.54971283596496024982127184848, 3.81048826503597000103626254713, 5.700782234257863227880996552816, 7.870642681763515302874401120649, 8.69780932428403701688834145613, 9.78494942273188896441817232531, 10.67277888180225591423109041140, 12.42127443733824031061475260068, 13.349014917555317218961801752159, 15.39804070309503314814008888612, 16.16890274728759033386371692009, 16.978751128723105275414836809682, 18.69506601402850364891367040220, 19.60689884626151795725917252345, 20.64374801142031887714347543873, 21.22830498316948477067568618387, 22.63534775371028283898610516638, 24.45403579160856096479175656987, 25.53922843262809031087462227219, 26.15340978277747899509512023358, 27.4631556321203814408411542868, 28.125714769132750197926930162415, 29.11843394879554617793219890136, 30.33312698682742850928070224920