| L(s) = 1 | + 2.35·2-s + 3.53·4-s + 3.60·5-s + 0.184·7-s + 3.60·8-s + 8.47·10-s − 3.16·11-s + 6.06·13-s + 0.434·14-s + 1.41·16-s − 3.31·17-s + 12.7·20-s − 7.45·22-s + 4.42·23-s + 7.98·25-s + 14.2·26-s + 0.652·28-s − 7.58·29-s + 7.41·31-s − 3.88·32-s − 7.80·34-s + 0.665·35-s + 1.77·37-s + 12.9·40-s + 2.21·41-s + 6.75·43-s − 11.1·44-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s + 1.61·5-s + 0.0698·7-s + 1.27·8-s + 2.68·10-s − 0.955·11-s + 1.68·13-s + 0.116·14-s + 0.352·16-s − 0.805·17-s + 2.84·20-s − 1.58·22-s + 0.921·23-s + 1.59·25-s + 2.79·26-s + 0.123·28-s − 1.40·29-s + 1.33·31-s − 0.687·32-s − 1.33·34-s + 0.112·35-s + 0.291·37-s + 2.05·40-s + 0.346·41-s + 1.03·43-s − 1.68·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.849264950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.849264950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.184T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 + 0.150T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 5.92T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 + 7.33T + 89T^{2} \) |
| 97 | \( 1 - 0.837T + 97T^{2} \) |
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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
−8.774599654356245368307607684370, −7.66144766352107653193801446411, −6.63759645988496897881304535917, −6.11830050139473281563464601611, −5.62033543246634226748486236168, −4.92569878875591844249857964906, −4.09503317921208224896722969649, −3.04090914177057804795829174580, −2.39506453392769439120762062558, −1.43433642977848946139503889083,
1.43433642977848946139503889083, 2.39506453392769439120762062558, 3.04090914177057804795829174580, 4.09503317921208224896722969649, 4.92569878875591844249857964906, 5.62033543246634226748486236168, 6.11830050139473281563464601611, 6.63759645988496897881304535917, 7.66144766352107653193801446411, 8.774599654356245368307607684370
