L(s) = 1 | + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (0.999 − 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (0.162 + 0.986i)37-s + ⋯ |
L(s) = 1 | + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (0.999 − 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (0.162 + 0.986i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.808131412 + 2.117989804i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808131412 + 2.117989804i\) |
\(L(1)\) |
\(\approx\) |
\(1.285500702 + 0.4413203858i\) |
\(L(1)\) |
\(\approx\) |
\(1.285500702 + 0.4413203858i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.227 + 0.973i)T \) |
| 5 | \( 1 + (0.582 - 0.812i)T \) |
| 11 | \( 1 + (0.999 - 0.0327i)T \) |
| 13 | \( 1 + (0.0980 + 0.995i)T \) |
| 17 | \( 1 + (0.793 + 0.608i)T \) |
| 19 | \( 1 + (-0.352 - 0.935i)T \) |
| 23 | \( 1 + (0.946 + 0.321i)T \) |
| 29 | \( 1 + (-0.881 + 0.471i)T \) |
| 31 | \( 1 + (-0.258 + 0.965i)T \) |
| 37 | \( 1 + (0.162 + 0.986i)T \) |
| 41 | \( 1 + (0.980 - 0.195i)T \) |
| 43 | \( 1 + (0.956 - 0.290i)T \) |
| 47 | \( 1 + (-0.991 - 0.130i)T \) |
| 53 | \( 1 + (0.0327 + 0.999i)T \) |
| 59 | \( 1 + (0.910 - 0.412i)T \) |
| 61 | \( 1 + (0.683 + 0.729i)T \) |
| 67 | \( 1 + (-0.973 + 0.227i)T \) |
| 71 | \( 1 + (0.831 - 0.555i)T \) |
| 73 | \( 1 + (-0.997 - 0.0654i)T \) |
| 79 | \( 1 + (-0.608 - 0.793i)T \) |
| 83 | \( 1 + (-0.773 - 0.634i)T \) |
| 89 | \( 1 + (0.751 + 0.659i)T \) |
| 97 | \( 1 + (-0.707 - 0.707i)T \) |
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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
−19.57920588190107851737851951500, −19.02520080494831716020471402050, −18.4113411241454322419304634220, −17.69853074255215147654674191399, −17.12363070223480525428354204734, −16.307525708370730586559707013854, −14.89129973542410556660203523211, −14.65620723224559949973481391183, −13.96499223113961123621698403603, −13.03029192115829600664473253081, −12.58690493263229635692152399839, −11.51139328016840673531006270805, −11.003561846555359869203739949365, −9.87597290251824464463265592351, −9.30805636643552216321116352963, −8.24863187645811985267224412014, −7.49327873227274572777674386666, −6.86893263959759130612085653209, −5.934754636360166780886314315863, −5.567440572437937579310178465851, −3.95640412148861012680918557372, −3.13159442380856555003650988946, −2.359999244493930857377151945059, −1.44497145515035103179933670415, −0.514478168383688850549633270089,
1.02328473591134281646018737292, 1.88174839536052520390743769447, 3.03899718416217876516947566147, 3.98637083313083072469595727136, 4.62652688138313147950472854794, 5.42956049296507401273637287650, 6.23035982843344595637854622049, 7.211667444148974147882054112586, 8.45645918557104515211160134529, 9.06648103378958417413342347937, 9.40888960887577878026592440272, 10.3521942284071302700955647626, 11.186457108834152713190995835482, 11.91059104865020930558307978577, 12.85334244320368423364776189222, 13.64397908646472809600200429158, 14.43043608645864274038582513436, 14.91007260989184150820799642801, 15.985931639968828332219226852182, 16.563254766136823500991360062012, 17.10941555497104808765590636898, 17.69531719688333146857022287893, 19.06852423630321022426514320172, 19.52246272560949770605309413535, 20.37833700725338396506119858237