| L(s) = 1 | + (1.53 − 1.28i)2-s + (0.694 − 3.93i)4-s + (−13.2 + 4.82i)5-s + (3.00 + 17.0i)7-s + (−4.00 − 6.92i)8-s + (−14.1 + 24.4i)10-s + (52.0 + 18.9i)11-s + (68.7 + 57.6i)13-s + (26.4 + 22.2i)14-s + (−15.0 − 5.47i)16-s + (−21.4 + 37.0i)17-s + (−39.9 − 69.1i)19-s + (9.79 + 55.5i)20-s + (104. − 37.8i)22-s + (−15.5 + 88.2i)23-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.18 + 0.431i)5-s + (0.162 + 0.919i)7-s + (−0.176 − 0.306i)8-s + (−0.445 + 0.772i)10-s + (1.42 + 0.519i)11-s + (1.46 + 1.23i)13-s + (0.505 + 0.424i)14-s + (−0.234 − 0.0855i)16-s + (−0.305 + 0.529i)17-s + (−0.481 − 0.834i)19-s + (0.109 + 0.621i)20-s + (1.00 − 0.367i)22-s + (−0.141 + 0.799i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.71415 + 0.662508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71415 + 0.662508i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 + 1.28i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (13.2 - 4.82i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-3.00 - 17.0i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-52.0 - 18.9i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-68.7 - 57.6i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (21.4 - 37.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (39.9 + 69.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (15.5 - 88.2i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (141. - 119. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-25.7 + 146. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (87.5 - 151. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-65.5 - 55.0i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-12.3 - 4.50i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-44.0 - 250. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 - 268.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-173. + 63.0i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (103. + 589. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-374. - 314. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-181. + 314. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (339. + 587. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (597. - 500. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-500. + 420. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (474. + 822. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-264. - 96.3i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.26162882487227882353747601566, −11.44644280094452133917111375030, −11.14730427290062910343620083156, −9.366202186741744822472816297641, −8.590367314231250966718635167103, −7.03223209205788364565838408859, −6.09669500746045235521499919827, −4.35675250282827975284728858350, −3.58843146760380996191020227673, −1.75253879542975837513565345317,
0.795084993145201893814246307677, 3.67708086691217406610891238998, 4.13837024115601177121881732776, 5.80248154028697593551181229482, 6.98603279663340722362279413070, 8.092037252740887918783707234173, 8.766919468270006734685470559104, 10.56586424971612376308858110566, 11.44301982560621675226338463887, 12.32581593996340563175393540916
