L(s) = 1 | + (0.707 + 0.707i)5-s + i·7-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + 17-s + (−0.707 + 0.707i)19-s − i·23-s + i·25-s + (−0.707 + 0.707i)29-s + 31-s + (−0.707 + 0.707i)35-s + (−0.707 − 0.707i)37-s − i·41-s + (0.707 + 0.707i)43-s + 47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s + i·7-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + 17-s + (−0.707 + 0.707i)19-s − i·23-s + i·25-s + (−0.707 + 0.707i)29-s + 31-s + (−0.707 + 0.707i)35-s + (−0.707 − 0.707i)37-s − i·41-s + (0.707 + 0.707i)43-s + 47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9648175001 + 1.175633525i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9648175001 + 1.175633525i\) |
\(L(1)\) |
\(\approx\) |
\(1.037047072 + 0.3977862730i\) |
\(L(1)\) |
\(\approx\) |
\(1.037047072 + 0.3977862730i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.707 + 0.707i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.707 + 0.707i)T \) |
| 61 | \( 1 + (0.707 - 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + T \) |
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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
−29.631026959405783107467904113585, −28.58295342044434504544986683121, −27.64497990067759501677561052631, −26.35608625473330342167845775280, −25.47922975971426750393724210909, −24.35877238835662353464923587412, −23.38919209493296263007175305092, −22.27182291005911326485834773427, −20.827970234844865582547928388233, −20.40346420885468122508773012006, −19.00507172188109317200681969160, −17.45663095646813714451077180289, −17.04748723300061775558249053502, −15.631440990995737388369720108872, −14.29078769588238132703768325045, −13.173451142504524051990278116079, −12.34622489996258700446846059346, −10.505241523279143754695982066956, −9.820212179307148950508770476667, −8.27032142840661838681767410892, −7.065212666372232272672515731970, −5.4643497165012661788790223054, −4.36427024933083804245328397739, −2.42359928932113692134862530247, −0.66836824823783880605269360371,
1.98806698935310155849984104960, 3.21401345039426940087606737065, 5.28877304309692124524815057806, 6.2221296589816990829602890439, 7.72791828215196771422872318732, 9.16522178574578139933632837627, 10.237531056184387963008915056555, 11.50039242745869136867271677604, 12.73206473066072292821565355031, 14.06179460747018106452466748357, 14.919633056726641169313050084393, 16.21657870661996581507932456051, 17.456797528858309709581569984632, 18.64262313991719246779269057982, 19.17399034747747946844149848887, 21.166752131518055774254678849810, 21.54157193056800171129809062105, 22.71485409274994313959488474853, 23.97332808585899609039542242040, 25.12316221417358690285077896122, 25.92831490680604368435959265660, 26.98513172616604793702112885686, 28.203449527312524821927095976575, 29.28069630917444538882474840806, 29.95041386620259439173626883496