L(s) = 1 | − i·2-s − i·3-s − 4-s − 0.500i·5-s − 6-s + i·8-s − 9-s − 0.500·10-s + (0.278 − 3.30i)11-s + i·12-s + 0.920·13-s − 0.500·15-s + 16-s − 3.39·17-s + i·18-s + 6.51·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.223i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.158·10-s + (0.0839 − 0.996i)11-s + 0.288i·12-s + 0.255·13-s − 0.129·15-s + 0.250·16-s − 0.822·17-s + 0.235i·18-s + 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.278583037\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.278583037\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.278 + 3.30i)T \) |
good | 5 | \( 1 + 0.500iT - 5T^{2} \) |
| 13 | \( 1 - 0.920T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 + 0.975T + 23T^{2} \) |
| 29 | \( 1 + 4.69iT - 29T^{2} \) |
| 31 | \( 1 + 8.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.56T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 + 0.666iT - 43T^{2} \) |
| 47 | \( 1 - 5.52iT - 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 11.7iT - 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 + 2.31T + 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 + 8.66iT - 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−8.244584991798200476220077524039, −7.73489321471681781258041252027, −6.70424021895158426767238256761, −5.94136541013586541139419983484, −5.20007626917750063294356493274, −4.21763652615703215351611755221, −3.30605662520097730386593669168, −2.51570251347989236508228159195, −1.39023750373119655171229581646, −0.42227291173326918108786031255,
1.40216808242143372616757624638, 2.83332975890215439289559301198, 3.67380387820299722194358927184, 4.67138669514501345770133324063, 5.13435525084116077659844696337, 6.04483527250907142896666756687, 7.08153953673031982887557460720, 7.22129899549754969710021599882, 8.462841489942930300817465629713, 8.935924262268579103210910630981