L(s) = 1 | + (0.692 − 0.721i)2-s + (−0.200 − 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.845 − 0.534i)6-s + (0.948 + 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (−0.970 + 0.239i)12-s + (−0.845 + 0.534i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (−0.996 + 0.0804i)16-s + (0.885 + 0.464i)17-s + ⋯ |
L(s) = 1 | + (0.692 − 0.721i)2-s + (−0.200 − 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.845 − 0.534i)6-s + (0.948 + 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (−0.970 + 0.239i)12-s + (−0.845 + 0.534i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (−0.996 + 0.0804i)16-s + (0.885 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4409559662 - 1.028376086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4409559662 - 1.028376086i\) |
\(L(1)\) |
\(\approx\) |
\(0.8429379693 - 0.8481669066i\) |
\(L(1)\) |
\(\approx\) |
\(0.8429379693 - 0.8481669066i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (0.692 - 0.721i)T \) |
| 3 | \( 1 + (-0.200 - 0.979i)T \) |
| 5 | \( 1 + (-0.996 - 0.0804i)T \) |
| 7 | \( 1 + (0.948 + 0.316i)T \) |
| 11 | \( 1 + (0.428 - 0.903i)T \) |
| 13 | \( 1 + (-0.845 + 0.534i)T \) |
| 17 | \( 1 + (0.885 + 0.464i)T \) |
| 19 | \( 1 + (0.987 - 0.160i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.799 - 0.600i)T \) |
| 31 | \( 1 + (0.278 + 0.960i)T \) |
| 37 | \( 1 + (-0.632 - 0.774i)T \) |
| 41 | \( 1 + (0.568 - 0.822i)T \) |
| 43 | \( 1 + (0.428 + 0.903i)T \) |
| 47 | \( 1 + (-0.632 + 0.774i)T \) |
| 53 | \( 1 + (-0.200 + 0.979i)T \) |
| 59 | \( 1 + (-0.0402 + 0.999i)T \) |
| 61 | \( 1 + (-0.354 + 0.935i)T \) |
| 67 | \( 1 + (-0.970 + 0.239i)T \) |
| 71 | \( 1 + (-0.748 - 0.663i)T \) |
| 73 | \( 1 + (-0.845 - 0.534i)T \) |
| 83 | \( 1 + (-0.0402 - 0.999i)T \) |
| 89 | \( 1 + (-0.748 + 0.663i)T \) |
| 97 | \( 1 + (-0.354 + 0.935i)T \) |
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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
−31.60234573046014778173041199698, −30.87963173743320706260999607847, −29.70524355037927344714868407580, −27.75931160231464373940792602677, −27.27758850048327624563659895091, −26.28570656271178925464084327839, −24.991452259529085752452451452155, −23.7772371199433913532793189990, −22.89525178405718509680442284669, −22.09210909336089717974211620925, −20.76674104184895718852829307123, −20.02147789349156221247391673184, −17.87492054282222986416734628205, −16.92269944199611119502450576411, −15.79929904885389312027281959562, −14.927126563056735140517581617121, −14.18154568172215496169336571980, −12.15430588839749434335458627233, −11.512049440983456245298442717544, −9.83990311914595515734868510314, −8.16083996144526598846281765978, −7.240673324439791048812877536247, −5.286399937034310814842515018446, −4.46717429566661469509537265378, −3.27450801892356215011944204527,
1.21693449347865705201792520500, 2.88792004467878319222504409278, 4.553600776197009970350031613100, 5.89347621321117521753353734742, 7.45365354483290944391659592000, 8.73450410848447085142511697595, 10.80448454689575015988842281721, 11.91017170501810709238917487224, 12.23416549469694924576591416809, 13.962310610262470126168311979646, 14.62303646959732751208144076440, 16.25819972901581838587888906508, 17.872657216783163629549564050488, 19.07820188240244554210640055385, 19.5836049472812685465100917897, 20.94480232631704033353857283168, 22.177751971220690253547895183025, 23.24627653693888434793431359199, 24.28529092882957967012823111557, 24.56064315247580819046848580316, 26.7882004907783136570019282385, 27.847971584067447896069590095696, 28.7752470556416063044726936134, 29.94793045401204421774483555339, 30.70049473989492146912334690126