L(s) = 1 | + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.974 − 0.222i)8-s + (0.0747 + 0.997i)11-s + (−0.997 + 0.0747i)13-s + (−0.222 + 0.974i)16-s + (0.930 − 0.365i)17-s + (0.5 + 0.866i)19-s + (−0.930 − 0.365i)22-s + (−0.149 − 0.988i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s + 31-s + (−0.781 − 0.623i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
L(s) = 1 | + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.974 − 0.222i)8-s + (0.0747 + 0.997i)11-s + (−0.997 + 0.0747i)13-s + (−0.222 + 0.974i)16-s + (0.930 − 0.365i)17-s + (0.5 + 0.866i)19-s + (−0.930 − 0.365i)22-s + (−0.149 − 0.988i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s + 31-s + (−0.781 − 0.623i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.305860293 + 0.4768339659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.305860293 + 0.4768339659i\) |
\(L(1)\) |
\(\approx\) |
\(0.7550681746 + 0.3065177166i\) |
\(L(1)\) |
\(\approx\) |
\(0.7550681746 + 0.3065177166i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.433 + 0.900i)T \) |
| 11 | \( 1 + (0.0747 + 0.997i)T \) |
| 13 | \( 1 + (-0.997 + 0.0747i)T \) |
| 17 | \( 1 + (0.930 - 0.365i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.149 - 0.988i)T \) |
| 29 | \( 1 + (-0.365 - 0.930i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.149 + 0.988i)T \) |
| 41 | \( 1 + (-0.733 + 0.680i)T \) |
| 43 | \( 1 + (0.680 - 0.733i)T \) |
| 47 | \( 1 + (-0.433 + 0.900i)T \) |
| 53 | \( 1 + (-0.149 - 0.988i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.997 + 0.0747i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.997 + 0.0747i)T \) |
| 89 | \( 1 + (-0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−19.46477909460010746062710860597, −18.97332907925689141396253721366, −18.07480137254502822941275187838, −17.45634334154145017490296739108, −16.74727556008057137455646166677, −16.11569773743389642538289540481, −15.09564013683091781605548921674, −14.13546459717719241583818152697, −13.61568581665299608419485339782, −12.74812328335541124201610586129, −12.03579552571106709690127478151, −11.42915284687714192255921673852, −10.65415276149754536983208649329, −9.92922789678501404007784049759, −9.22828024096457348508219006130, −8.50848221771786859402164405267, −7.66064697923125169003949787458, −7.03541345252911038956249063732, −5.68498675684792934815043890209, −5.05044090839194344786628517117, −3.96610307342960969820299368092, −3.21291024537207697646233092171, −2.5201890429508789381126940062, −1.39994372908466496953661725310, −0.60097373197798934422182256328,
0.46479210641329148860620735252, 1.52939122352769302206631019705, 2.50660870370524083325561509233, 3.7917036086871595393590080304, 4.76105521281747045016353147988, 5.23581074747282652420546925977, 6.320408250236229591193748341599, 6.91304329144591086410671020688, 7.85551513508244604052861659533, 8.1529496310826233621839520748, 9.50335224539531398061752936394, 9.7890155296408655051086980121, 10.43988543467863296030188210851, 11.70154082859427164722158995435, 12.343079748490409061645149678134, 13.19049360029752673973659241969, 14.28378363127238162859719231584, 14.48110963165188045528738940992, 15.38997003544369354975519610148, 16.006985335822983834489071104957, 17.02804357151478758238515555167, 17.141733077988913352618311300173, 18.18769358504189442229619578107, 18.760573653158648578659021805048, 19.40097608988312554528285532532