L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)13-s + (0.913 + 0.406i)14-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.978 + 0.207i)22-s + (0.913 + 0.406i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.669 + 0.743i)29-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)13-s + (0.913 + 0.406i)14-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.978 + 0.207i)22-s + (0.913 + 0.406i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.669 + 0.743i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8776843362 + 0.01225558160i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8776843362 + 0.01225558160i\) |
\(L(1)\) |
\(\approx\) |
\(0.8434325677 - 0.1837787997i\) |
\(L(1)\) |
\(\approx\) |
\(0.8434325677 - 0.1837787997i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.104 + 0.994i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 23 | \( 1 + (0.913 + 0.406i)T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.104 + 0.994i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.669 - 0.743i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.978 - 0.207i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.669 + 0.743i)T \) |
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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
−26.39084032154950906521504826584, −25.398776501017351982960014398357, −24.81153747540903238168274593416, −23.53577959343937664576876468779, −22.925017664920852198524391879874, −22.242064112129079990524642413629, −20.706195340519784016701910416935, −19.8209045808272081479224734162, −18.72812351669497915339762575564, −17.598739331993836880373237470480, −17.09887928061570297161435787724, −15.84442411640335024061815458159, −15.2607397126821907010940674419, −14.00120071052519378898778210872, −13.27288810421951421321488454166, −12.22625438977108239662703904117, −10.47062321399941101980635915712, −9.79465378782660962466591234374, −8.60913993761493754544118596423, −7.3080950318822704052004771529, −6.87443860030248954036183046545, −5.34962657744728325487376708194, −4.45714385688126599863530785167, −3.029653839061093913544585912742, −0.74733003715179359807946303918,
1.461055700823681773862009861628, 2.81849529970840476759143468286, 3.7794371522878066930036445014, 5.22094027233739061876406040636, 6.31190164056410015171942766446, 8.08502487095384246699921782888, 8.957499843195380767759231023471, 9.86445961564035036190413106948, 11.013815954709816227599572252315, 11.9254123637277729394146397761, 12.760675592893390969895199866266, 13.79786875181724684266386343731, 14.78003320635344927397492577761, 16.17985925713370053930106830737, 17.06428052224519914740739581842, 18.37851226160956676327612664846, 19.024524258372821050173313765240, 19.648763290075385811767160022898, 21.183146404869135742606322311454, 21.45911566500590700253063983552, 22.49675680707815245124448008400, 23.453133735185483743602994044842, 24.54167645095715262071410228574, 25.751256852305585986439460882185, 26.60011719173165502606460134293