| L(s) = 1 | − 3.56·5-s − 3.12·7-s + 5.12·13-s + 2·17-s + 4·19-s − 2.43·23-s + 7.68·25-s − 5.12·29-s − 5.56·31-s + 11.1·35-s − 7.56·37-s − 1.12·41-s + 7.12·43-s − 8·47-s + 2.75·49-s − 12.2·53-s − 7.80·59-s − 1.12·61-s − 18.2·65-s + 9.56·67-s + 8.68·71-s − 5.12·73-s + 11.1·79-s + 0.876·83-s − 7.12·85-s − 2.68·89-s − 16·91-s + ⋯ |
| L(s) = 1 | − 1.59·5-s − 1.18·7-s + 1.42·13-s + 0.485·17-s + 0.917·19-s − 0.508·23-s + 1.53·25-s − 0.951·29-s − 0.998·31-s + 1.88·35-s − 1.24·37-s − 0.175·41-s + 1.08·43-s − 1.16·47-s + 0.393·49-s − 1.68·53-s − 1.01·59-s − 0.143·61-s − 2.26·65-s + 1.16·67-s + 1.03·71-s − 0.599·73-s + 1.25·79-s + 0.0962·83-s − 0.772·85-s − 0.284·89-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7994848498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7994848498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.83759616300998346646168961788, −7.12442818796937239745216254711, −6.48405108424065690282867098688, −5.77503212644068543761214247225, −4.95346993736919533949863942038, −3.86629603531077187530636678049, −3.57948479042599992143172610233, −3.06936253573987295863097995127, −1.59139422094954619345736616727, −0.44146815386578294920970869460,
0.44146815386578294920970869460, 1.59139422094954619345736616727, 3.06936253573987295863097995127, 3.57948479042599992143172610233, 3.86629603531077187530636678049, 4.95346993736919533949863942038, 5.77503212644068543761214247225, 6.48405108424065690282867098688, 7.12442818796937239745216254711, 7.83759616300998346646168961788
