L(s) = 1 | + (−0.564 − 0.825i)2-s + (−0.809 − 0.587i)3-s + (−0.362 + 0.931i)4-s + (−0.921 + 0.389i)5-s + (−0.0285 + 0.999i)6-s + (−0.870 − 0.491i)7-s + (0.974 − 0.226i)8-s + (0.309 + 0.951i)9-s + (0.841 + 0.540i)10-s + (0.841 − 0.540i)12-s + (0.774 + 0.633i)13-s + (0.0855 + 0.996i)14-s + (0.974 + 0.226i)15-s + (−0.736 − 0.676i)16-s + (0.897 + 0.441i)17-s + (0.610 − 0.791i)18-s + ⋯ |
L(s) = 1 | + (−0.564 − 0.825i)2-s + (−0.809 − 0.587i)3-s + (−0.362 + 0.931i)4-s + (−0.921 + 0.389i)5-s + (−0.0285 + 0.999i)6-s + (−0.870 − 0.491i)7-s + (0.974 − 0.226i)8-s + (0.309 + 0.951i)9-s + (0.841 + 0.540i)10-s + (0.841 − 0.540i)12-s + (0.774 + 0.633i)13-s + (0.0855 + 0.996i)14-s + (0.974 + 0.226i)15-s + (−0.736 − 0.676i)16-s + (0.897 + 0.441i)17-s + (0.610 − 0.791i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3844587951 + 0.009984168580i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3844587951 + 0.009984168580i\) |
\(L(1)\) |
\(\approx\) |
\(0.4670954296 - 0.1316608842i\) |
\(L(1)\) |
\(\approx\) |
\(0.4670954296 - 0.1316608842i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.564 - 0.825i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.921 + 0.389i)T \) |
| 7 | \( 1 + (-0.870 - 0.491i)T \) |
| 13 | \( 1 + (0.774 + 0.633i)T \) |
| 17 | \( 1 + (0.897 + 0.441i)T \) |
| 19 | \( 1 + (-0.466 + 0.884i)T \) |
| 23 | \( 1 + (0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.696 + 0.717i)T \) |
| 31 | \( 1 + (-0.254 + 0.967i)T \) |
| 37 | \( 1 + (-0.998 + 0.0570i)T \) |
| 41 | \( 1 + (0.941 + 0.336i)T \) |
| 43 | \( 1 + (-0.654 + 0.755i)T \) |
| 47 | \( 1 + (0.610 + 0.791i)T \) |
| 53 | \( 1 + (-0.736 + 0.676i)T \) |
| 59 | \( 1 + (0.941 - 0.336i)T \) |
| 61 | \( 1 + (-0.564 + 0.825i)T \) |
| 67 | \( 1 + (-0.959 + 0.281i)T \) |
| 71 | \( 1 + (-0.985 - 0.170i)T \) |
| 73 | \( 1 + (0.198 + 0.980i)T \) |
| 79 | \( 1 + (0.516 - 0.856i)T \) |
| 83 | \( 1 + (0.993 - 0.113i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.921 - 0.389i)T \) |
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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
−28.47266897197024886237749021153, −27.95385244409829390190442565812, −27.1998412499986457092609642518, −26.09205120530790876707910564453, −25.134759700612702716237950798248, −23.81406927286320390254488376888, −23.12439271593625761296813317637, −22.37091236177450790081080175701, −20.84926639905584775953851495419, −19.542093663994259128706126519937, −18.64776896617089414607085487281, −17.44846471780019101080306381379, −16.445826342894038911829983522786, −15.67133620006349517533520618382, −15.16200500321783779835451821623, −13.26880587974071827655510348402, −11.97099067960474302021643137052, −10.81334997249204191600503477956, −9.6268058259239288062518631860, −8.65722441789715498198503519566, −7.26164852761393499934139467986, −6.02620934351280393026857587779, −5.04479282468580884699353816511, −3.6154628954756435559725914337, −0.57821782563149301902579621083,
1.233767449489422916390852384444, 3.15799934973019870910267171126, 4.33550571537569061933391078010, 6.40498039228349992500777387919, 7.4030072800382034087614182562, 8.559530728113507993804275931744, 10.29349698896810138658765363718, 10.920968841674624265143188378935, 12.14822718003555128527840133691, 12.73996544689225909702991115825, 14.11957674746274798451035586342, 16.17879382997270259120254100122, 16.62233366690479316469311616717, 18.00692876292226461642833573844, 19.0245299780314891760878457874, 19.36536421924571929131340588354, 20.76610874760663031087850416680, 22.0539800405457116412229870829, 23.06066679587700510317600980735, 23.49578800532772439056865295189, 25.25512371784092467170513467905, 26.28112841546144609464855237586, 27.28714128816844913158377140509, 28.17844496212075642612500332414, 29.04010613041875416815088452621