L(s) = 1 | + (−2.5 − 4.33i)5-s + (−1 + 1.73i)7-s + (−15 + 25.9i)11-s + (2 + 3.46i)13-s + 90·17-s − 28·19-s + (−60 − 103. i)23-s + (−12.5 + 21.6i)25-s + (−105 + 181. i)29-s + (2 + 3.46i)31-s + 10·35-s + 200·37-s + (−120 − 207. i)41-s + (68 − 117. i)43-s + (60 − 103. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.0539 + 0.0935i)7-s + (−0.411 + 0.712i)11-s + (0.0426 + 0.0739i)13-s + 1.28·17-s − 0.338·19-s + (−0.543 − 0.942i)23-s + (−0.100 + 0.173i)25-s + (−0.672 + 1.16i)29-s + (0.0115 + 0.0200i)31-s + 0.0482·35-s + 0.888·37-s + (−0.457 − 0.791i)41-s + (0.241 − 0.417i)43-s + (0.186 − 0.322i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3331622265\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3331622265\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 7 | \( 1 + (1 - 1.73i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 90T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + (60 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (105 - 181. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 200T + 5.06e4T^{2} \) |
| 41 | \( 1 + (120 + 207. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-68 + 117. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-60 + 103. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 30T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-225 - 389. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-83 + 143. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (454 + 786. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 250T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-458 + 793. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-570 + 987. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 420T + 7.04e5T^{2} \) |
| 97 | \( 1 + (769 - 1.33e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−8.713671906088203592311723777997, −7.80947908342359151375510984630, −7.25866805029369186443985096867, −6.17528960484236701451022299210, −5.34890602703545337268806780692, −4.52610248911436309000694764837, −3.60785198718358548573447097410, −2.48946374736915228736623506539, −1.37232588779270444710161745246, −0.07681495809115050831755858804,
1.17473545460130699301810812103, 2.53295497382072620116038362480, 3.42904024109421979011817073283, 4.24115095677107864646307847472, 5.52172674261808539951620271367, 5.99225580246452438874083777557, 7.09163031409123613086511297252, 7.86927649750114272249377465992, 8.368239522967900127809999200971, 9.577975142715556788500628192157