| L(s) = 1 | + 4i·2-s − 16·4-s − 47i·7-s − 64i·8-s − 222·11-s + 101i·13-s + 188·14-s + 256·16-s + 162i·17-s − 1.68e3·19-s − 888i·22-s − 306i·23-s − 404·26-s + 752i·28-s + 7.89e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.362i·7-s − 0.353i·8-s − 0.553·11-s + 0.165i·13-s + 0.256·14-s + 0.250·16-s + 0.135i·17-s − 1.07·19-s − 0.391i·22-s − 0.120i·23-s − 0.117·26-s + 0.181i·28-s + 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.580317815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580317815\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 47iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 222T + 1.61e5T^{2} \) |
| 13 | \( 1 - 101iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 162iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.68e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 306iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 7.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.64e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.43e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.09e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.79e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 107iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.47e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 823iT - 8.58e9T^{2} \) |
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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
−10.43524768640312594995690423099, −9.409752495331592442034511686926, −8.499062591484156708715608330817, −7.66848001324565883663412994068, −6.75867316304925232062080097300, −5.84689711890490573158909254234, −4.76168897673030163512005715634, −3.81616559935190045537131504160, −2.33773059429520049498204790659, −0.71150664222530705688436511650,
0.57378078154384954054164338439, 2.01282817183255031494802720666, 2.96536391751812507813815874103, 4.20313586494754919300679895789, 5.21927576015611910427867213164, 6.25783472320988400852693215564, 7.52763011083938784428574055157, 8.519512774047546782163230570283, 9.255362625122993155293963565153, 10.36572198410571433259244319562
