| L(s) = 1 | + (−0.5 − 1.65i)3-s − 2·4-s + (−2.5 + 1.65i)9-s + 3.31i·11-s + (1 + 3.31i)12-s + 4·16-s + 3.31i·23-s + (4 + 3.31i)27-s + 5·31-s + (5.5 − 1.65i)33-s + (5 − 3.31i)36-s + 7·37-s − 6.63i·44-s − 6.63i·47-s + (−2 − 6.63i)48-s + 7·49-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.957i)3-s − 4-s + (−0.833 + 0.552i)9-s + 1.00i·11-s + (0.288 + 0.957i)12-s + 16-s + 0.691i·23-s + (0.769 + 0.638i)27-s + 0.898·31-s + (0.957 − 0.288i)33-s + (0.833 − 0.552i)36-s + 1.15·37-s − 1.00i·44-s − 0.967i·47-s + (−0.288 − 0.957i)48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.886491 + 0.130736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.886491 + 0.130736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.5 + 1.65i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−10.14227301099561192935192207698, −9.409182833037618912498027650738, −8.459582344309918943582826979578, −7.72141310655749135853035834097, −6.90488283908595489963992609625, −5.83309292293605728561818711920, −5.00510841172561509148863106667, −3.99933567035749169194434985543, −2.51980887316730286192218929707, −1.09043292891394823764737667832,
0.59348566073184814548066850810, 2.94980270813203111496784793472, 3.93847797538450776196369128125, 4.73572244239777490345529975486, 5.61802711583685558357014286463, 6.41876903374681373226512089844, 7.994039951037752938719613530357, 8.635040350410288143573535924694, 9.385806550711685090966418171406, 10.11008616430074033542740960533
