| L(s) = 1 | + 178.·3-s + 872. i·5-s + 1.24e4i·7-s + 1.22e4·9-s + 2.03e4i·11-s + (6.57e4 + 7.92e4i)13-s + 1.56e5i·15-s + 1.82e5·17-s − 3.54e5i·19-s + 2.22e6i·21-s + 1.23e6·23-s + 1.19e6·25-s − 1.32e6·27-s + 4.17e6·29-s + 3.16e6i·31-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + 0.624i·5-s + 1.95i·7-s + 0.624·9-s + 0.418i·11-s + (0.638 + 0.769i)13-s + 0.795i·15-s + 0.529·17-s − 0.624i·19-s + 2.49i·21-s + 0.916·23-s + 0.610·25-s − 0.478·27-s + 1.09·29-s + 0.615i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.552723739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.552723739\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-6.57e4 - 7.92e4i)T \) |
| good | 3 | \( 1 - 178.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 872. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 1.24e4iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 2.03e4iT - 2.35e9T^{2} \) |
| 17 | \( 1 - 1.82e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.54e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 1.23e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.16e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.27e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 7.40e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.44e5iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 9.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.05e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 1.66e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.19e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 5.98e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 3.96e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 2.20e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.19e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 6.79e7iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 1.66e9iT - 7.60e17T^{2} \) |
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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
−11.17098957953929969147183327075, −9.800627884417736035807059759363, −8.852237170871430578767598071157, −8.542840399662288868466401663929, −7.16107096299243274366766008126, −6.09991718370241709254106532544, −4.82006101415533277019934957216, −3.18135900148236390702938149463, −2.67333123283150632379198019126, −1.64115604518609281708497631848,
0.62967299964326229082282394317, 1.36890099415317771589517649161, 3.10561127365348251837898107070, 3.75596864704710024980304913087, 4.91276924527411720303271468514, 6.52324268753184967610816252826, 7.79858882972761355699506471742, 8.186240312273793809521382483956, 9.335778126624933414932968723142, 10.29274575469746495539141729356
