L(s) = 1 | + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (0.116 + 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (−0.448 + 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (0.116 + 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (−0.448 + 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233026249 + 1.589740402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233026249 + 1.589740402i\) |
\(L(1)\) |
\(\approx\) |
\(1.347990645 + 0.2113206100i\) |
\(L(1)\) |
\(\approx\) |
\(1.347990645 + 0.2113206100i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.835 + 0.549i)T \) |
| 5 | \( 1 + (0.116 + 0.993i)T \) |
| 7 | \( 1 + (0.597 + 0.802i)T \) |
| 11 | \( 1 + (-0.998 + 0.0581i)T \) |
| 13 | \( 1 + (0.597 + 0.802i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.549 + 0.835i)T \) |
| 31 | \( 1 + (0.918 - 0.396i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.642 + 0.766i)T \) |
| 47 | \( 1 + (0.957 + 0.286i)T \) |
| 53 | \( 1 + (-0.918 - 0.396i)T \) |
| 59 | \( 1 + (0.893 + 0.448i)T \) |
| 61 | \( 1 + (0.230 - 0.973i)T \) |
| 67 | \( 1 + (-0.116 + 0.993i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.0581 + 0.998i)T \) |
| 79 | \( 1 + (0.597 - 0.802i)T \) |
| 83 | \( 1 + (0.835 + 0.549i)T \) |
| 89 | \( 1 + (-0.230 + 0.973i)T \) |
| 97 | \( 1 + (0.116 + 0.993i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.98313617772389387817417653298, −17.54145416582070015492429111224, −17.113305761250878366532008794953, −15.93557440700726594091981937772, −15.58168568743864844864128061112, −15.03887756367263847370107105036, −13.89673097051223295712334252922, −13.54494837798544516717538507845, −13.22088220717411506087340998107, −12.487868420353892545990174728981, −11.605972112842823196321537890827, −10.46155322726051274401956015436, −9.74020431486181453548553190497, −8.8486782960828229393003722733, −8.40693042457926576456794011999, −7.721805419300014430862668956467, −7.381279076223104975949539159129, −6.34207236520006610981494579527, −5.52256538490231913530074267198, −4.89288021454001765124152549984, −4.02081003831125910117740772048, −3.44135074091469093386690612898, −2.27127658471034416102669009932, −1.2297680064992277022487694747, −0.470930821226979299848975033617,
1.4781896651266599320657153274, 2.280833419342691007929871423375, 2.703576773949897853439733787093, 3.399725928823650108120981679304, 4.309416594781351450821606318844, 4.93999295707572839925206857661, 5.72729359004087909449401894228, 6.71193692704528661469769738584, 7.86448496712793564003188989598, 8.31762666482566455312302078312, 9.18074505712019525887272821345, 9.71770183549783145752172269207, 10.45090152601655703441728203379, 11.10789772155635302297450738261, 11.53495781615244673576586277543, 12.49316539702318508200006617768, 13.39615060833996685595205814731, 13.866517551494783908972683687123, 14.53196794482009010783094914943, 14.99515119648286461219184073673, 15.76013459177636572299434498153, 16.43538568261392640393144385625, 17.84321270411124458555081037659, 18.18865321961583600905446808750, 18.82790745429122802457671102186