L(s) = 1 | + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.104 − 0.994i)5-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.809 − 0.587i)20-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.104 − 0.994i)5-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.809 − 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9442463037 - 0.6988240891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9442463037 - 0.6988240891i\) |
\(L(1)\) |
\(\approx\) |
\(1.145030259 - 0.4984948103i\) |
\(L(1)\) |
\(\approx\) |
\(1.145030259 - 0.4984948103i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (-0.104 + 0.994i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.669 - 0.743i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−31.42933606140196346461261269949, −30.29610265405904493749719002058, −29.685932271714357201175259891468, −28.57414744185050955869543147746, −27.021992048998316892867509018714, −26.05037964550580471691783358642, −24.649418940570576832465766440114, −23.7816011759878900136158063233, −22.675363067161471006302246352999, −22.177620080289219564629280179505, −21.048179863264765909209071256346, −19.30432021414415948322428523890, −18.05475189581830823524962273656, −16.92534873997691337898076941141, −15.8657410680657203936570854753, −14.66974020257679530764687178063, −13.57659734434689735545106341451, −12.114431459794336638568072070251, −11.426217896338131745970801797986, −10.08286069603019642299812490505, −7.60785484137363936929046827256, −6.77279216658514603429071913346, −5.61658870394791246801907551998, −4.2608404478461868492501779950, −2.52494644260805032222452064832,
1.342856223193901756105551633436, 3.67407271847625370832755303873, 5.05093947737247259122393243910, 5.73908944736511454230547600460, 7.46801752958991146004760676139, 9.62162917153890874125358155635, 10.69449573325807594986526744791, 12.272155974815066278146776938851, 12.399734516165618043136858977981, 14.05881170721534582972681778596, 15.55622856835363975994299013425, 16.40142717744921663321117456409, 17.59408679068100705174551659813, 19.22067600919081805956371587854, 20.424165549923272974406697626333, 21.37204068580564691611698166245, 22.354030028295980456723273760742, 23.4194356914808698674278493097, 24.1782217905649304909178096730, 25.21902605858544259622130303211, 27.32550836928007389029831517038, 28.02171568619077138431870778504, 29.10783989587114791941692530679, 29.724528935882130083650445678479, 31.172566138919632863802131052335