L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.499 − 0.866i)12-s − 13-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.499 − 0.866i)12-s − 13-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.777109093\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.777109093\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.350894455665634825915813175140, −8.777503927959084304691888426820, −7.87701851748890129326974159971, −6.77372900700427541987131161141, −5.38041955204161873916832420171, −4.94005786471534710057054977478, −4.36923170371027214131957889596, −3.33005458699690209835047358318, −2.39283543949854310280213563058, −1.15378852336528667087010971805,
1.89178482420119405851070783966, 2.88577167346002804629276132508, 3.53244402960607043832521683221, 5.19062005555007256658895226144, 5.54491691264746875952153422219, 6.58921217996425815774353985199, 7.11487224481875368759826745235, 7.918156656559265513221592671703, 8.394637039021529159142520994155, 9.559903733794370035359536627609