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
−29.95041386620259439173626883496, −29.28069630917444538882474840806, −28.203449527312524821927095976575, −26.98513172616604793702112885686, −25.92831490680604368435959265660, −25.12316221417358690285077896122, −23.97332808585899609039542242040, −22.71485409274994313959488474853, −21.54157193056800171129809062105, −21.166752131518055774254678849810, −19.17399034747747946844149848887, −18.64262313991719246779269057982, −17.456797528858309709581569984632, −16.21657870661996581507932456051, −14.919633056726641169313050084393, −14.06179460747018106452466748357, −12.73206473066072292821565355031, −11.50039242745869136867271677604, −10.237531056184387963008915056555, −9.16522178574578139933632837627, −7.72791828215196771422872318732, −6.2221296589816990829602890439, −5.28877304309692124524815057806, −3.21401345039426940087606737065, −1.98806698935310155849984104960,
0.66836824823783880605269360371, 2.42359928932113692134862530247, 4.36427024933083804245328397739, 5.4643497165012661788790223054, 7.065212666372232272672515731970, 8.27032142840661838681767410892, 9.820212179307148950508770476667, 10.505241523279143754695982066956, 12.34622489996258700446846059346, 13.173451142504524051990278116079, 14.29078769588238132703768325045, 15.631440990995737388369720108872, 17.04748723300061775558249053502, 17.45663095646813714451077180289, 19.00507172188109317200681969160, 20.40346420885468122508773012006, 20.827970234844865582547928388233, 22.27182291005911326485834773427, 23.38919209493296263007175305092, 24.35877238835662353464923587412, 25.47922975971426750393724210909, 26.35608625473330342167845775280, 27.64497990067759501677561052631, 28.58295342044434504544986683121, 29.631026959405783107467904113585