L(s) = 1 | + 11.4·3-s + (−32.3 − 45.6i)5-s + 121. i·7-s − 113.·9-s − 625. i·11-s + 924.·13-s + (−368. − 520. i)15-s + 1.32e3i·17-s + 1.13e3i·19-s + 1.38e3i·21-s − 3.05e3i·23-s + (−1.03e3 + 2.94e3i)25-s − 4.05e3·27-s − 8.23e3i·29-s − 1.98e3·31-s + ⋯ |
L(s) = 1 | + 0.731·3-s + (−0.578 − 0.816i)5-s + 0.936i·7-s − 0.465·9-s − 1.55i·11-s + 1.51·13-s + (−0.422 − 0.596i)15-s + 1.11i·17-s + 0.721i·19-s + 0.684i·21-s − 1.20i·23-s + (−0.331 + 0.943i)25-s − 1.07·27-s − 1.81i·29-s − 0.371·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7883050859\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7883050859\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (32.3 + 45.6i)T \) |
good | 3 | \( 1 - 11.4T + 243T^{2} \) |
| 7 | \( 1 - 121. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 625. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 924.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.13e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.05e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 8.23e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.58e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.87e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.16e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.86e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.41e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.53e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.88e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.52e4iT - 8.58e9T^{2} \) |
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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
−10.43210335854101103871283118165, −8.887584908047294096800063385147, −8.500089995534653214343494234747, −8.155553634911458339107456496440, −6.18460365693317381702400533037, −5.59417916939813869680181469504, −3.94859048453074893437890501539, −3.18214385628907600116073625047, −1.67162516279474129472334902464, −0.17956546767667700984454589762,
1.60701752802672699830252883358, 3.09961688106245020911820321762, 3.78461868174468609333562515600, 5.10762189486011206432403119945, 6.85710779596742302874161152701, 7.23978114835976594330805205513, 8.330491767193192137116892771246, 9.304630472654388940177434745178, 10.33354187026775710983209185965, 11.15524678693381428830139726924