L(s) = 1 | + 1.41i·2-s − i·3-s + (0.224 + 2.22i)5-s + 1.41·6-s + 1.73i·7-s + 2.82i·8-s − 9-s + (−3.14 + 0.317i)10-s + 6.29i·13-s − 2.44·14-s + (2.22 − 0.224i)15-s − 4.00·16-s − 2.04i·17-s − 1.41i·18-s − 1.73·19-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 0.577i·3-s + (0.100 + 0.994i)5-s + 0.577·6-s + 0.654i·7-s + 0.999i·8-s − 0.333·9-s + (−0.994 + 0.100i)10-s + 1.74i·13-s − 0.654·14-s + (0.574 − 0.0580i)15-s − 1.00·16-s − 0.497i·17-s − 0.333i·18-s − 0.397·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.478735907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478735907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.224 - 2.22i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.41iT - 2T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 13 | \( 1 - 6.29iT - 13T^{2} \) |
| 17 | \( 1 + 2.04iT - 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 + 3.89iT - 37T^{2} \) |
| 41 | \( 1 - 2.19T + 41T^{2} \) |
| 43 | \( 1 + 5.65iT - 43T^{2} \) |
| 47 | \( 1 + 7.34iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−9.293827209035269354146894733404, −8.812337400457492465503066093197, −7.77628116230317201170794076396, −7.15661657989102346569605415551, −6.57413895663002701794098046152, −6.02660871178717852995381432162, −5.14300436615255347770604853053, −3.90402935671172400495731823744, −2.50427074126901049129634797628, −2.01013625963307612993191875221,
0.51457974833832237274760943081, 1.59048223297747150126521059862, 2.89011194626489841263249353849, 3.72528023738358523613633165705, 4.48643402222868496093574621170, 5.45243376895937187689253435576, 6.26237075056535629437133461072, 7.53386919001246126719004635206, 8.167834089258285527517110769344, 9.130041338184679436583121388538