L(s) = 1 | − 2.35·3-s + (−2.66 + 2.66i)7-s + 2.56·9-s + (2.20 + 2.20i)11-s + 4.16i·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 + 1.03·27-s + (3.65 − 3.65i)29-s + 6.90i·31-s + (−5.20 − 5.20i)33-s + 1.10i·37-s − 9.81i·39-s + ⋯ |
L(s) = 1 | − 1.36·3-s + (−1.00 + 1.00i)7-s + 0.853·9-s + (0.665 + 0.665i)11-s + 1.15i·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 + 0.199·27-s + (0.679 − 0.679i)29-s + 1.23i·31-s + (−0.905 − 0.905i)33-s + 0.180i·37-s − 1.57i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1548843948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1548843948\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.35T + 3T^{2} \) |
| 7 | \( 1 + (2.66 - 2.66i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.20 - 2.20i)T + 11iT^{2} \) |
| 13 | \( 1 - 4.16iT - 13T^{2} \) |
| 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 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 9.85iT - 41T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 + (3.90 + 3.90i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + (3.42 - 3.42i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.57 + 4.57i)T + 61iT^{2} \) |
| 67 | \( 1 + 6.37iT - 67T^{2} \) |
| 71 | \( 1 - 1.03T + 71T^{2} \) |
| 73 | \( 1 + (4.70 - 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 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
−9.844419076199860719026845683893, −9.169655554904686641046027010936, −8.558012427309920384743735224935, −6.91241218810877545073936849239, −6.64674287656647633694477319966, −5.99376855468535572974465916884, −4.97058982285656633828044253713, −4.31277061357395281378501599372, −2.94381401482238991601649831626, −1.68548294783599849848704380882,
0.087950168534269794961537814875, 1.01722456124191365546097857271, 2.94725447143285613580748989930, 3.93756043398100421834845780048, 4.80893553420891464974443280101, 5.93297731448263270160507700420, 6.31564072666969987810359822403, 7.04600029884221805350590991472, 8.048430083694529867994610871146, 9.029168830119732216231143914761