L(s) = 1 | + (−0.730 + 0.730i)3-s + (1.08 + 2.41i)7-s + 1.93i·9-s + 5.95·11-s + (−0.921 + 0.921i)13-s + (2.02 + 2.02i)17-s − 3.29·19-s + (−2.55 − 0.970i)21-s + (−0.0544 − 0.0544i)23-s + (−3.60 − 3.60i)27-s − 3.78i·29-s − 4.88i·31-s + (−4.35 + 4.35i)33-s + (3.20 − 3.20i)37-s − 1.34i·39-s + ⋯ |
L(s) = 1 | + (−0.421 + 0.421i)3-s + (0.409 + 0.912i)7-s + 0.644i·9-s + 1.79·11-s + (−0.255 + 0.255i)13-s + (0.491 + 0.491i)17-s − 0.755·19-s + (−0.557 − 0.211i)21-s + (−0.0113 − 0.0113i)23-s + (−0.693 − 0.693i)27-s − 0.703i·29-s − 0.876i·31-s + (−0.757 + 0.757i)33-s + (0.527 − 0.527i)37-s − 0.215i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.522245020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522245020\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.08 - 2.41i)T \) |
good | 3 | \( 1 + (0.730 - 0.730i)T - 3iT^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + (0.921 - 0.921i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.02 - 2.02i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + (0.0544 + 0.0544i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.78iT - 29T^{2} \) |
| 31 | \( 1 + 4.88iT - 31T^{2} \) |
| 37 | \( 1 + (-3.20 + 3.20i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (-1.60 - 1.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.64 - 2.64i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.16 - 9.16i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 + (-6.43 + 6.43i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + (8.66 - 8.66i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.76iT - 79T^{2} \) |
| 83 | \( 1 + (2.36 - 2.36i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.73T + 89T^{2} \) |
| 97 | \( 1 + (2.05 + 2.05i)T + 97iT^{2} \) |
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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
−9.685601751455967210868676728581, −9.091527480274476796530694070114, −8.265638550380895184344549320772, −7.42411983219891458674432143635, −6.16503664019264020808684390850, −5.85147173353025009605717543941, −4.57165581064011850162258661967, −4.10744820797369401961595297865, −2.59877574848729709680867756448, −1.50043547967924141122873746653,
0.71709467187575896033437505297, 1.67642062167987909715050000387, 3.41478927426336973007722069419, 4.11402043308266832354453232540, 5.16723519382958415828496236963, 6.26361456147383174643338986382, 6.87488866998700189018414732014, 7.46511904837380423200708347825, 8.640312119643614775674666954512, 9.265347959783695396168549789569