| L(s) = 1 | − 1.50·3-s − 3.53·5-s − 4.04·7-s − 0.729·9-s − 13-s + 5.32·15-s + 3.02·17-s + 0.120·19-s + 6.09·21-s − 5.56·23-s + 7.47·25-s + 5.61·27-s − 6.19·29-s + 6.22·31-s + 14.2·35-s − 2.36·37-s + 1.50·39-s + 8.86·41-s + 2.93·43-s + 2.57·45-s − 13.4·47-s + 9.34·49-s − 4.55·51-s + 4.66·53-s − 0.182·57-s + 6.19·59-s − 0.351·61-s + ⋯ |
| L(s) = 1 | − 0.869·3-s − 1.57·5-s − 1.52·7-s − 0.243·9-s − 0.277·13-s + 1.37·15-s + 0.733·17-s + 0.0277·19-s + 1.32·21-s − 1.16·23-s + 1.49·25-s + 1.08·27-s − 1.15·29-s + 1.11·31-s + 2.41·35-s − 0.388·37-s + 0.241·39-s + 1.38·41-s + 0.447·43-s + 0.384·45-s − 1.95·47-s + 1.33·49-s − 0.638·51-s + 0.640·53-s − 0.0241·57-s + 0.806·59-s − 0.0449·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 3.53T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 - 0.120T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 - 4.66T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 + 0.351T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 6.27T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.71068146834673731486714646075, −6.87759388105591574309386993322, −6.32296893757142961210147361321, −5.64346917603177669320425852317, −4.81470605402136571344979997162, −3.85976105310860847869398083309, −3.45321932787780098653021541773, −2.53014004670678674376057998842, −0.73946101109720186885899243146, 0,
0.73946101109720186885899243146, 2.53014004670678674376057998842, 3.45321932787780098653021541773, 3.85976105310860847869398083309, 4.81470605402136571344979997162, 5.64346917603177669320425852317, 6.32296893757142961210147361321, 6.87759388105591574309386993322, 7.71068146834673731486714646075
