L(s) = 1 | + (0.944 − 0.327i)2-s + (−0.283 − 0.959i)3-s + (0.785 − 0.619i)4-s + (0.914 − 0.405i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (0.538 − 0.842i)8-s + (−0.839 + 0.543i)9-s + (0.730 − 0.682i)10-s + (0.797 − 0.602i)11-s + (−0.816 − 0.577i)12-s + (−0.998 + 0.0521i)13-s + (−0.999 − 0.0313i)14-s + (−0.647 − 0.761i)15-s + (0.232 − 0.972i)16-s + (−0.995 + 0.0937i)17-s + ⋯ |
L(s) = 1 | + (0.944 − 0.327i)2-s + (−0.283 − 0.959i)3-s + (0.785 − 0.619i)4-s + (0.914 − 0.405i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (0.538 − 0.842i)8-s + (−0.839 + 0.543i)9-s + (0.730 − 0.682i)10-s + (0.797 − 0.602i)11-s + (−0.816 − 0.577i)12-s + (−0.998 + 0.0521i)13-s + (−0.999 − 0.0313i)14-s + (−0.647 − 0.761i)15-s + (0.232 − 0.972i)16-s + (−0.995 + 0.0937i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4889393010 - 1.754844974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4889393010 - 1.754844974i\) |
\(L(1)\) |
\(\approx\) |
\(1.023175819 - 1.098834765i\) |
\(L(1)\) |
\(\approx\) |
\(1.023175819 - 1.098834765i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.944 - 0.327i)T \) |
| 3 | \( 1 + (-0.283 - 0.959i)T \) |
| 5 | \( 1 + (0.914 - 0.405i)T \) |
| 7 | \( 1 + (-0.934 - 0.357i)T \) |
| 11 | \( 1 + (0.797 - 0.602i)T \) |
| 13 | \( 1 + (-0.998 + 0.0521i)T \) |
| 17 | \( 1 + (-0.995 + 0.0937i)T \) |
| 19 | \( 1 + (-0.900 + 0.433i)T \) |
| 23 | \( 1 + (-0.381 - 0.924i)T \) |
| 29 | \( 1 + (0.716 - 0.697i)T \) |
| 31 | \( 1 + (-0.663 + 0.748i)T \) |
| 37 | \( 1 + (0.391 + 0.920i)T \) |
| 41 | \( 1 + (-0.948 - 0.317i)T \) |
| 47 | \( 1 + (-0.839 - 0.543i)T \) |
| 53 | \( 1 + (0.538 + 0.842i)T \) |
| 59 | \( 1 + (-0.971 + 0.237i)T \) |
| 61 | \( 1 + (0.556 - 0.831i)T \) |
| 67 | \( 1 + (-0.529 - 0.848i)T \) |
| 71 | \( 1 + (-0.0989 + 0.995i)T \) |
| 73 | \( 1 + (-0.438 - 0.898i)T \) |
| 79 | \( 1 + (0.957 + 0.288i)T \) |
| 83 | \( 1 + (0.556 - 0.831i)T \) |
| 89 | \( 1 + (-0.918 - 0.395i)T \) |
| 97 | \( 1 + (0.171 + 0.985i)T \) |
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
−20.85729520771191554252606966436, −19.81854237625068497580055005396, −19.58957145702067807067488142505, −17.958586693205537926885935421413, −17.432576799016588836881363958875, −16.750901000484071669721233638123, −16.10247929817292392614215902426, −15.08831416097158220443282675589, −14.9346810180754046952937969325, −14.04616066350223682277056717985, −13.186573216795557398052573426705, −12.51164055106212431286919240484, −11.68127971951887605363997741459, −10.91753980811717053217583744265, −10.030998250810499441110880792454, −9.43504126445395274863066331783, −8.69988254449782542939345403782, −7.18966281710958158937512898769, −6.57853124465290314065193502175, −5.97306468788027618553204151461, −5.15161189087843953278375067207, −4.40574846988104440179667815362, −3.5506933678519777306401440452, −2.673481044330457752470539127253, −1.989441096979453830193462967560,
0.40795649415746229462140725054, 1.55688657090131282946444407910, 2.28321691356685051232149557197, 3.0773248091837640121239100267, 4.259814223601324377229344448544, 5.04693242573748769546851146371, 6.13646278001381922109861416168, 6.406103308886987572400879697661, 7.021682937317368911922115981135, 8.3150514191849393701519805973, 9.22820376135792330298739648790, 10.19064154838072275968110147209, 10.760480453976851899085331925, 11.891644121971524816884581951413, 12.37374550690678435262450797957, 13.06002363360038599700387739912, 13.62387092605756538796413920786, 14.18752441717815209738179754317, 14.982525218148808451734023553617, 16.217663641260401159637218804523, 16.81590229337912114348656997608, 17.28816916169079150070625665388, 18.4247739683525866661574844780, 19.149401473313665567018636962122, 19.86739516856877736503296864222