L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4542820890 - 0.4596636635i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4542820890 - 0.4596636635i\) |
\(L(1)\) |
\(\approx\) |
\(0.7124830147 - 0.4910059593i\) |
\(L(1)\) |
\(\approx\) |
\(0.7124830147 - 0.4910059593i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.841 + 0.540i)T \) |
| 17 | \( 1 + (-0.959 - 0.281i)T \) |
| 19 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 + 0.909i)T \) |
| 37 | \( 1 + (-0.654 - 0.755i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.841 + 0.540i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.959 + 0.281i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−39.630589253427970962995821234711, −37.77411630115861677116390700405, −36.83136652728095244607274574675, −35.31986669431009589084467665916, −34.187775530292962937658311082520, −32.87424424000170917884878324688, −31.87412179805184831834560608843, −30.97805367598638152476017439155, −28.1210607618274055065119894514, −27.50777201752062610708812593831, −26.275631990238364246164021111486, −24.83033504983248348613401784470, −23.76188687440743738536230743016, −22.066231640167673219479742483795, −20.67494430032699797550929546084, −19.00505634356831335793236361495, −17.16891944881079233650210363882, −15.85253243312657114363761589162, −15.04251051590667695399815539530, −13.380213450669045687542426834143, −11.00692064811433854564371405694, −8.828560229821020532492420431612, −8.23455744973127504023626840065, −5.527258524941725784466998943930, −4.12168134698006821351736110537,
2.044762235917227441005602451891, 4.03528117503497395606199794164, 7.178383719762419099693442656676, 8.580891341384805742436590726156, 10.75645338483036295494981316768, 11.94789351008338603970116213884, 13.52059537639149946355024865231, 14.77864637462672441075329062038, 17.5871153342747464320750956621, 18.55761841830637896742463096719, 19.77526761173815985849828230239, 20.87208035387172726679522644384, 22.84970431975498903297629724134, 23.75888055708372168717217522032, 25.83869469482731532443275203768, 26.96277899701958235711638522869, 28.42778028807417555980305445107, 30.00548137405077635256440342745, 30.65281967237935792977330619243, 31.51461797195130987311578820832, 33.71450450824251765414541776951, 35.405010925966572274206402222, 36.23026982523501020290933788759, 37.5173950391724365292793859954, 38.50340307558800926550590898790