L(s) = 1 | − 3.91i·5-s + (−2.03 + 1.68i)7-s − 1.62·11-s + 2.54·13-s + 2.94·17-s − 2.72·19-s + 8.57i·23-s − 10.3·25-s + 8.21·29-s + 8.08i·31-s + (6.61 + 7.98i)35-s + 8.05i·37-s + 9.17·41-s − 9.05i·43-s − 1.17·47-s + ⋯ |
L(s) = 1 | − 1.75i·5-s + (−0.769 + 0.638i)7-s − 0.490·11-s + 0.706·13-s + 0.713·17-s − 0.624·19-s + 1.78i·23-s − 2.07·25-s + 1.52·29-s + 1.45i·31-s + (1.11 + 1.34i)35-s + 1.32i·37-s + 1.43·41-s − 1.38i·43-s − 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.580465138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.580465138\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.03 - 1.68i)T \) |
good | 5 | \( 1 + 3.91iT - 5T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 - 8.08iT - 31T^{2} \) |
| 37 | \( 1 - 8.05iT - 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 9.05iT - 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 1.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 + 7.35iT - 67T^{2} \) |
| 71 | \( 1 + 7.71iT - 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 0.0913T + 79T^{2} \) |
| 83 | \( 1 - 2.03iT - 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 7.53iT - 97T^{2} \) |
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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
−8.476824052420292656941086510684, −7.997808909458323587861023643455, −6.91664591569284399806492191321, −5.99514584832954132266814797796, −5.41150905311425032301105479708, −4.82656994518539099350252942491, −3.83443500012250102714445413020, −3.00984237164333138431759619496, −1.73493857306204468398452054660, −0.822609854577796003625352833063,
0.63129777762781364599854800162, 2.39752574002889338078569293596, 2.87784512330334478784883588209, 3.77876259832574416444933211339, 4.43913632106623269759856829958, 5.91165442149419650890204647804, 6.29795095902966035357376595103, 6.90311887353714127465315397599, 7.65551589147206238030172242957, 8.247783068196387070801227343863