L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s − 1.77i·11-s + (−0.692 + 0.692i)13-s + 1.00·14-s − 1.00·16-s + (−2.39 + 2.39i)17-s + (1.25 + 1.25i)22-s + (3.38 + 3.38i)23-s − 0.979i·26-s + (−0.707 + 0.707i)28-s + 4.42·29-s − 5.77·31-s + (0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s − 0.536i·11-s + (−0.192 + 0.192i)13-s + 0.267·14-s − 0.250·16-s + (−0.580 + 0.580i)17-s + (0.268 + 0.268i)22-s + (0.705 + 0.705i)23-s − 0.192i·26-s + (−0.133 + 0.133i)28-s + 0.821·29-s − 1.03·31-s + (0.125 − 0.125i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8283488324\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8283488324\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 1.77iT - 11T^{2} \) |
| 13 | \( 1 + (0.692 - 0.692i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.39 - 2.39i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-3.38 - 3.38i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + (5.91 + 5.91i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.807iT - 41T^{2} \) |
| 43 | \( 1 + (-4.64 + 4.64i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.47 - 7.47i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.56 - 3.56i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + (-0.641 - 0.641i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 + (5.70 - 5.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 16.3iT - 79T^{2} \) |
| 83 | \( 1 + (0.171 + 0.171i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + (12.6 + 12.6i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.834739666744061995435576412230, −8.270130907458926871251101285243, −7.29194332971047641637046074440, −6.86940488870952465529325487940, −5.94931584399218244993680362340, −5.31095783076481907880738184930, −4.28204143024171523979981193867, −3.41228755982244574129217258589, −2.24079904223674948345022594074, −1.03762019743924534077857832254,
0.34724157076307811648912964621, 1.75261877867588580424357219594, 2.66484219710249802420293049454, 3.46681618673785245254679224481, 4.58044125887515865785113855032, 5.20425525635692083705114372139, 6.42031925943431416667988638743, 6.99376490626342140836333331994, 7.78230142340953683325279269597, 8.680857255839507685335354276107