L(s) = 1 | + 1.27·5-s + i·7-s + 2.84i·11-s − 0.223i·13-s − 0.397i·17-s + 4.64·19-s + 7.36·23-s − 3.36·25-s − 10.6·29-s + 7.67i·31-s + 1.27i·35-s + 4.74i·37-s + 1.97i·41-s − 7.62·43-s + 11.0·47-s + ⋯ |
L(s) = 1 | + 0.571·5-s + 0.377i·7-s + 0.858i·11-s − 0.0620i·13-s − 0.0964i·17-s + 1.06·19-s + 1.53·23-s − 0.673·25-s − 1.97·29-s + 1.37i·31-s + 0.216i·35-s + 0.780i·37-s + 0.308i·41-s − 1.16·43-s + 1.61·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.869450767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.869450767\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 11 | \( 1 - 2.84iT - 11T^{2} \) |
| 13 | \( 1 + 0.223iT - 13T^{2} \) |
| 17 | \( 1 + 0.397iT - 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 7.67iT - 31T^{2} \) |
| 37 | \( 1 - 4.74iT - 37T^{2} \) |
| 41 | \( 1 - 1.97iT - 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 0.295T + 53T^{2} \) |
| 59 | \( 1 - 7.25iT - 59T^{2} \) |
| 61 | \( 1 + 9.45iT - 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.22iT - 79T^{2} \) |
| 83 | \( 1 + 9.11iT - 83T^{2} \) |
| 89 | \( 1 - 8.94iT - 89T^{2} \) |
| 97 | \( 1 - 0.228T + 97T^{2} \) |
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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
−8.269247158136097520552189807818, −7.37786181425472547375048702740, −6.98882288033987509579984899152, −6.07132114695075466562728901553, −5.28000625568993648361005281980, −4.91980182728593635587036795060, −3.75886029911626036801294566968, −2.97662162675423155493943424882, −2.05196193220526394803705989888, −1.22182559670471593948504479392,
0.48294038218327591776266357351, 1.55571167130939036268456911600, 2.54409491628425415808495478554, 3.47999078104056876551494720272, 4.09185521496575901725560033395, 5.30316984254090898558194428936, 5.60927378147316654941612123760, 6.42061167586427800645270898163, 7.33545811464936561462257380801, 7.69022861213046416760166574841