L(s) = 1 | + 3·5-s + (−2 + 1.73i)7-s + 5.19i·11-s − 3.46i·13-s − 6·17-s + 1.73i·19-s − 5.19i·23-s + 4·25-s + 10.3i·29-s + 5.19i·31-s + (−6 + 5.19i)35-s + 37-s − 3·41-s − 10·43-s + 6·47-s + ⋯ |
L(s) = 1 | + 1.34·5-s + (−0.755 + 0.654i)7-s + 1.56i·11-s − 0.960i·13-s − 1.45·17-s + 0.397i·19-s − 1.08i·23-s + 0.800·25-s + 1.92i·29-s + 0.933i·31-s + (−1.01 + 0.878i)35-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.241958986\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.241958986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 5.19iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−8.925613923807683273725080869698, −8.610190444332558145458579339150, −7.21430312976100682884341875726, −6.71640111409935872663977329888, −5.98826615237472315433458688777, −5.20919025650180824820006485366, −4.53735762689570983327516331494, −3.16833700260265901994212883162, −2.37577536382199931415664425892, −1.61341195233487171607866973412,
0.34832893972256891121407140849, 1.74076340317028418083910345390, 2.64552566914310812396839325346, 3.65667840947889635022657714151, 4.50371149494191360826947695984, 5.61578839880169822821976620403, 6.28127445935905180683440789933, 6.61629398488388570737120462336, 7.66888029502150422230762208426, 8.657458522502953569830698386143