L(s) = 1 | + (−0.0572 + 1.41i)2-s + (−1.99 − 0.161i)4-s − 0.386i·5-s − i·7-s + (0.342 − 2.80i)8-s + (0.545 + 0.0221i)10-s − 3.49·11-s + 5.33·13-s + (1.41 + 0.0572i)14-s + (3.94 + 0.645i)16-s + 4.58i·17-s + 6.22i·19-s + (−0.0625 + 0.770i)20-s + (0.200 − 4.94i)22-s + 8.20·23-s + ⋯ |
L(s) = 1 | + (−0.0404 + 0.999i)2-s + (−0.996 − 0.0809i)4-s − 0.172i·5-s − 0.377i·7-s + (0.121 − 0.992i)8-s + (0.172 + 0.00699i)10-s − 1.05·11-s + 1.47·13-s + (0.377 + 0.0153i)14-s + (0.986 + 0.161i)16-s + 1.11i·17-s + 1.42i·19-s + (−0.0139 + 0.172i)20-s + (0.0426 − 1.05i)22-s + 1.71·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.978153 + 0.901971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.978153 + 0.901971i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0572 - 1.41i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.386iT - 5T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 - 4.58iT - 17T^{2} \) |
| 19 | \( 1 - 6.22iT - 19T^{2} \) |
| 23 | \( 1 - 8.20T + 23T^{2} \) |
| 29 | \( 1 - 6.76iT - 29T^{2} \) |
| 31 | \( 1 + 9.47iT - 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 - 4.43iT - 41T^{2} \) |
| 43 | \( 1 + 1.95iT - 43T^{2} \) |
| 47 | \( 1 - 0.996T + 47T^{2} \) |
| 53 | \( 1 + 1.92iT - 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 - 3.49iT - 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 8.14T + 73T^{2} \) |
| 79 | \( 1 - 12.0iT - 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 0.206iT - 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−10.53606521476044402650723733083, −9.528050894390118079323089316942, −8.509482778346942238948334344899, −8.079119999270722263297897179818, −7.07359889552364637242310493205, −6.11606134450274515569994399683, −5.40221419288997103799666875513, −4.28824895435096269112457408937, −3.32649129475675998127918438765, −1.19484768206430251439563490627,
0.883260058473794130346036914264, 2.58515673329587797106475976839, 3.21352404636799004202797041153, 4.68970440783702862122225728782, 5.32032768447419518697903832993, 6.63444568536070793049248498426, 7.74606674526403138837686172604, 8.852674970410966887683498328665, 9.148948467506740824283882398127, 10.42371375066831993995689386452