| L(s) = 1 | + 2.44i·2-s + 10·4-s + (5 + 24.4i)5-s + (−35 − 34.2i)7-s + 63.6i·8-s + (−59.9 + 12.2i)10-s − 89·11-s + 5·13-s + (84 − 85.7i)14-s + 4.00·16-s − 485·17-s − 220. i·19-s + (50 + 244. i)20-s − 218. i·22-s − 700. i·23-s + ⋯ |
| L(s) = 1 | + 0.612i·2-s + 0.625·4-s + (0.200 + 0.979i)5-s + (−0.714 − 0.699i)7-s + 0.995i·8-s + (−0.599 + 0.122i)10-s − 0.735·11-s + 0.0295·13-s + (0.428 − 0.437i)14-s + 0.0156·16-s − 1.67·17-s − 0.610i·19-s + (0.125 + 0.612i)20-s − 0.450i·22-s − 1.32i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1633779426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1633779426\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-5 - 24.4i)T \) |
| 7 | \( 1 + (35 + 34.2i)T \) |
| good | 2 | \( 1 - 2.44iT - 16T^{2} \) |
| 11 | \( 1 + 89T + 1.46e4T^{2} \) |
| 13 | \( 1 - 5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 485T + 8.35e4T^{2} \) |
| 19 | \( 1 + 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 700. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 191T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 377. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.19e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.58e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.04e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.65e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.99e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.23e3T + 8.85e7T^{2} \) |
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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
−11.22900491095701623564158164117, −10.82965063688310999118171381344, −9.952930220127865199619935731591, −8.634714194782791850035843018392, −7.48012815397376878824328982461, −6.73167918927131602249684325445, −6.22285384362553345742874667540, −4.73838229870557115932693650134, −3.15382063623107178931521874578, −2.21153654761809677650772367690,
0.04271474778270711330162631474, 1.67981434593764966618792505346, 2.68823585970890093825370399899, 4.03410771558254527636632476992, 5.45248054277781380323096382115, 6.31982248214292066746242620224, 7.54860526471867691137951902647, 8.748049791746956837393632211875, 9.555655290809640471052219670554, 10.40260692780998536605267491689
