L(s) = 1 | − 2.35i·3-s + (−2.66 + 2.66i)7-s − 2.56·9-s + (2.20 − 2.20i)11-s + 4.16·13-s + (−1.69 + 1.69i)17-s + (4.74 − 4.74i)19-s + (6.28 + 6.28i)21-s + (3.70 + 3.70i)23-s − 1.03i·27-s + (−3.65 − 3.65i)29-s − 6.90i·31-s + (−5.20 − 5.20i)33-s − 1.10·37-s − 9.81i·39-s + ⋯ |
L(s) = 1 | − 1.36i·3-s + (−1.00 + 1.00i)7-s − 0.853·9-s + (0.665 − 0.665i)11-s + 1.15·13-s + (−0.410 + 0.410i)17-s + (1.08 − 1.08i)19-s + (1.37 + 1.37i)21-s + (0.772 + 0.772i)23-s − 0.199i·27-s + (−0.679 − 0.679i)29-s − 1.23i·31-s + (−0.905 − 0.905i)33-s − 0.180·37-s − 1.57i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.508662257\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.508662257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.35iT - 3T^{2} \) |
| 7 | \( 1 + (2.66 - 2.66i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.20 + 2.20i)T - 11iT^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 + (1.69 - 1.69i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.74 + 4.74i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.70 - 3.70i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.65 + 3.65i)T + 29iT^{2} \) |
| 31 | \( 1 + 6.90iT - 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 + 9.85iT - 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + (-3.90 - 3.90i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.19iT - 53T^{2} \) |
| 59 | \( 1 + (-3.42 - 3.42i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.57 - 4.57i)T - 61iT^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 1.03T + 71T^{2} \) |
| 73 | \( 1 + (-4.70 + 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 + 7.65iT - 83T^{2} \) |
| 89 | \( 1 - 1.77T + 89T^{2} \) |
| 97 | \( 1 + (1.16 - 1.16i)T - 97iT^{2} \) |
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
−9.041744414773084788334513428273, −8.427207650381340183489017702009, −7.42715394263153519903887460822, −6.68400110993276736174008001359, −6.08340513119688052648354972417, −5.45833870813772354768130219622, −3.80563979090393463432949512140, −2.94628202454791097015825901818, −1.87321222224941118550387223954, −0.66149500550467328293041127244,
1.27539931361263054991845052742, 3.26634645250576068441012268990, 3.64714611411824810467305072097, 4.51639947321597889734878381845, 5.35821068460119061812554713575, 6.55837826040865609330734486504, 7.02636863017850831084113734423, 8.251586376173990381055082197184, 9.209450713790417944880646151852, 9.636704438720761264821044404962