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 | \( 1 + (12.6 - 12.6i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.680857255839507685335354276107, −7.78230142340953683325279269597, −6.99376490626342140836333331994, −6.42031925943431416667988638743, −5.20425525635692083705114372139, −4.58044125887515865785113855032, −3.46681618673785245254679224481, −2.66484219710249802420293049454, −1.75261877867588580424357219594, −0.34724157076307811648912964621,
1.03762019743924534077857832254, 2.24079904223674948345022594074, 3.41228755982244574129217258589, 4.28204143024171523979981193867, 5.31095783076481907880738184930, 5.94931584399218244993680362340, 6.86940488870952465529325487940, 7.29194332971047641637046074440, 8.270130907458926871251101285243, 8.834739666744061995435576412230