| L(s) = 1 | + (0.761 − 1.08i)2-s + (0.0804 + 0.221i)4-s + (−1.62 + 1.53i)5-s + (1.77 − 0.827i)7-s + (2.86 + 0.768i)8-s + (0.439 + 2.93i)10-s + (2.76 + 3.29i)11-s + (−4.15 + 2.90i)13-s + (0.451 − 2.56i)14-s + (2.66 − 2.23i)16-s + (3.09 − 0.828i)17-s + (0.776 − 0.448i)19-s + (−0.470 − 0.234i)20-s + (5.69 − 0.498i)22-s + (2.36 − 5.07i)23-s + ⋯ |
| L(s) = 1 | + (0.538 − 0.769i)2-s + (0.0402 + 0.110i)4-s + (−0.725 + 0.688i)5-s + (0.670 − 0.312i)7-s + (1.01 + 0.271i)8-s + (0.138 + 0.929i)10-s + (0.833 + 0.993i)11-s + (−1.15 + 0.806i)13-s + (0.120 − 0.684i)14-s + (0.665 − 0.558i)16-s + (0.749 − 0.200i)17-s + (0.178 − 0.102i)19-s + (−0.105 − 0.0524i)20-s + (1.21 − 0.106i)22-s + (0.493 − 1.05i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87696 + 0.00418324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87696 + 0.00418324i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.62 - 1.53i)T \) |
| good | 2 | \( 1 + (-0.761 + 1.08i)T + (-0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.77 + 0.827i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-2.76 - 3.29i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (4.15 - 2.90i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-3.09 + 0.828i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.776 + 0.448i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.36 + 5.07i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 6.50i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 0.828i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.86 + 6.96i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (8.33 + 1.47i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.28 - 0.112i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (3.48 + 7.46i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.947 - 0.947i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.72 + 3.12i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.90 + 2.87i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.53 - 7.90i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (4.92 + 2.84i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.32 - 4.93i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.410 + 0.0724i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.59 - 5.31i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (0.974 + 1.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.17 + 13.4i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−11.42057406254611170158856548322, −10.64015050818526952059538490858, −9.749878549960816889307692499035, −8.400372602369753322866756277169, −7.26157704633109886693900851470, −6.96302690645508525184199104311, −4.88102876215266760976867260808, −4.26024883097949169159790530827, −3.11998956543857143126901484553, −1.86211776254636970875108603205,
1.24516452207141572610305073045, 3.41396595991809683721517098788, 4.73166757576866744645879874538, 5.34735213761991456879672162445, 6.37461469461234889465027636517, 7.67676455735521877808469043392, 8.092374649482704118046105120101, 9.319335350853141991279872662819, 10.36768517913466118441216993842, 11.54812970173310553772284414026
