| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.23i·5-s − 3.49i·7-s + 2.82·8-s + 3.16i·10-s − 11.9i·11-s − 19.9·13-s − 4.94i·14-s + 4.00·16-s + 26.2i·17-s + 0.0297i·19-s + 4.47i·20-s − 16.9i·22-s + (9.54 + 20.9i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.447i·5-s − 0.499i·7-s + 0.353·8-s + 0.316i·10-s − 1.08i·11-s − 1.53·13-s − 0.353i·14-s + 0.250·16-s + 1.54i·17-s + 0.00156i·19-s + 0.223i·20-s − 0.769i·22-s + (0.414 + 0.909i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.081883694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.081883694\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-9.54 - 20.9i)T \) |
| good | 7 | \( 1 + 3.49iT - 49T^{2} \) |
| 11 | \( 1 + 11.9iT - 121T^{2} \) |
| 13 | \( 1 + 19.9T + 169T^{2} \) |
| 17 | \( 1 - 26.2iT - 289T^{2} \) |
| 19 | \( 1 - 0.0297iT - 361T^{2} \) |
| 29 | \( 1 - 41.0T + 841T^{2} \) |
| 31 | \( 1 - 29.7T + 961T^{2} \) |
| 37 | \( 1 - 51.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 74.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 27.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 15.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.55T + 3.48e3T^{2} \) |
| 61 | \( 1 + 12.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 28.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 94.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 25.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 55.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 81.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.03iT - 9.40e3T^{2} \) |
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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
−8.922358927196950282743178082373, −8.022737506496630943846192317118, −7.36445007017582520937890483629, −6.50721044849071528660411389749, −5.86783518200467216159614926727, −4.91510963127918536651981424218, −4.06484552764960906753796972615, −3.19328396130695935344537701254, −2.36115033345549045118181118402, −0.952413498929509031978042043196,
0.73440998258792875726594097113, 2.39807334597041127152684963145, 2.68485196876775348745321730977, 4.34301314169861434727392330223, 4.76236214004389531027369036762, 5.44965364008183809946962294491, 6.54760574736150632628197549952, 7.25344699465935043472924139897, 7.895435149529113705307978357511, 9.045812159578614548304278414464
