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 | \( 1 + (1.16 + 1.16i)T + 97iT^{2} \) |
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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
−9.636704438720761264821044404962, −9.209450713790417944880646151852, −8.251586376173990381055082197184, −7.02636863017850831084113734423, −6.55837826040865609330734486504, −5.35821068460119061812554713575, −4.51639947321597889734878381845, −3.64714611411824810467305072097, −3.26634645250576068441012268990, −1.27539931361263054991845052742,
0.66149500550467328293041127244, 1.87321222224941118550387223954, 2.94628202454791097015825901818, 3.80563979090393463432949512140, 5.45833870813772354768130219622, 6.08340513119688052648354972417, 6.68400110993276736174008001359, 7.42715394263153519903887460822, 8.427207650381340183489017702009, 9.041744414773084788334513428273