L(s) = 1 | + (0.877 − 0.479i)3-s + (0.947 − 0.319i)5-s + (0.958 + 0.283i)7-s + (0.539 − 0.842i)9-s + (0.452 − 0.891i)11-s + (0.883 + 0.468i)13-s + (0.677 − 0.735i)15-s + (−0.640 − 0.768i)17-s + (0.155 + 0.987i)19-s + (0.977 − 0.211i)21-s + (0.934 + 0.355i)23-s + (0.795 − 0.605i)25-s + (0.0687 − 0.997i)27-s + (−0.570 + 0.821i)29-s + (−0.787 + 0.615i)31-s + ⋯ |
L(s) = 1 | + (0.877 − 0.479i)3-s + (0.947 − 0.319i)5-s + (0.958 + 0.283i)7-s + (0.539 − 0.842i)9-s + (0.452 − 0.891i)11-s + (0.883 + 0.468i)13-s + (0.677 − 0.735i)15-s + (−0.640 − 0.768i)17-s + (0.155 + 0.987i)19-s + (0.977 − 0.211i)21-s + (0.934 + 0.355i)23-s + (0.795 − 0.605i)25-s + (0.0687 − 0.997i)27-s + (−0.570 + 0.821i)29-s + (−0.787 + 0.615i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.590037749 - 1.636172963i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.590037749 - 1.636172963i\) |
\(L(1)\) |
\(\approx\) |
\(1.970937427 - 0.4972817999i\) |
\(L(1)\) |
\(\approx\) |
\(1.970937427 - 0.4972817999i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (0.877 - 0.479i)T \) |
| 5 | \( 1 + (0.947 - 0.319i)T \) |
| 7 | \( 1 + (0.958 + 0.283i)T \) |
| 11 | \( 1 + (0.452 - 0.891i)T \) |
| 13 | \( 1 + (0.883 + 0.468i)T \) |
| 17 | \( 1 + (-0.640 - 0.768i)T \) |
| 19 | \( 1 + (0.155 + 0.987i)T \) |
| 23 | \( 1 + (0.934 + 0.355i)T \) |
| 29 | \( 1 + (-0.570 + 0.821i)T \) |
| 31 | \( 1 + (-0.787 + 0.615i)T \) |
| 37 | \( 1 + (0.217 - 0.976i)T \) |
| 41 | \( 1 + (-0.474 + 0.880i)T \) |
| 43 | \( 1 + (-0.871 - 0.490i)T \) |
| 47 | \( 1 + (0.0562 - 0.998i)T \) |
| 53 | \( 1 + (-0.155 + 0.987i)T \) |
| 59 | \( 1 + (0.192 - 0.981i)T \) |
| 61 | \( 1 + (0.0312 - 0.999i)T \) |
| 67 | \( 1 + (0.677 + 0.735i)T \) |
| 71 | \( 1 + (0.852 - 0.523i)T \) |
| 73 | \( 1 + (0.910 + 0.412i)T \) |
| 79 | \( 1 + (-0.764 + 0.644i)T \) |
| 83 | \( 1 + (-0.998 + 0.0625i)T \) |
| 89 | \( 1 + (-0.920 + 0.389i)T \) |
| 97 | \( 1 + (-0.905 + 0.424i)T \) |
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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
−18.43307258553240039036908522783, −17.924253604359847278526229765511, −17.18365208958754083269411862450, −16.74106096079877019174448789922, −15.44367574862863859774101980523, −15.128043950958622963939014206471, −14.59112763379247563509471567101, −13.79308241437299179331945759944, −13.27177154229324275645501747553, −12.753617056200203554936329442032, −11.32172686611499789754872262244, −10.989940097165725059345540033734, −10.179304199091759522338350846468, −9.58248161814821053157256892983, −8.81823853440435631995913296931, −8.32279152061543506949731331892, −7.355695196379280300099616230268, −6.77955983557847687655280870491, −5.79352675253138894331582033629, −4.921811883112746342072935369733, −4.34420357209892001450311277571, −3.5145674561852119255886925948, −2.57265915242955977567922256048, −1.89967232850026457915025223479, −1.25490646281247701948697175063,
1.06830171425523753839872016608, 1.58463371062908159735369504265, 2.248935401275188418737647428531, 3.25754727162912787527109474933, 3.915786667025858019739966555630, 5.02970822243792151411671861644, 5.62741992190124686092677769577, 6.52377991173804970191096336466, 7.11969536645220882289218777279, 8.14954966473757952815637991336, 8.73600572508404764715333658175, 9.089397447102956474483113974960, 9.85365312788642369793182769583, 10.99256041653979289746385247538, 11.39247691128711467020534462853, 12.41530921769733524257444884750, 13.01987529874275124873060043267, 13.90318649025542178980520714223, 14.02227928873809665575513241942, 14.73878534486539053129059844244, 15.58074216948515008892965526953, 16.44165495216926113349277822200, 17.028251740831939601569784997055, 17.94102294995306575674094519797, 18.49940453351806123070181474667