L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s − 0.999·12-s + (−1 − 3.46i)13-s + 1.99·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.204 + 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.150 + 0.261i)11-s − 0.288·12-s + (−0.277 − 0.960i)13-s + 0.534·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.552238 - 0.545202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.552238 - 0.545202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.90937670186892991577168890429, −10.12517065453462735269260901831, −9.034459442604326415178631316709, −8.207630922051620251699860032955, −7.24143500776294464532185579915, −6.75291627981381086354924312477, −5.37145907856310411810976535529, −4.15062477912618303856347626263, −2.65892728883515571680451632700, −0.54642108553476561140389230166,
2.05450076650254382549536794668, 3.38417984286778871633795605843, 4.31115519860193369590440338129, 5.65399534387303613384059772834, 6.97485854802334134421766797877, 8.221564931204832666314638305541, 8.862285601515102236885771913953, 9.730605789479653896164563439402, 10.57789869681260290506921332746, 11.49797621387494279074194656732