| L(s) = 1 | − 2·7-s − 2·11-s − 6·13-s + 17-s − 19-s − 23-s + 5·29-s + 8·31-s + 3·37-s − 10·41-s − 2·43-s − 7·47-s − 3·49-s − 5·59-s + 8·61-s − 67-s + 12·71-s − 11·73-s + 4·77-s − 8·79-s − 12·83-s − 11·89-s + 12·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s − 1.66·13-s + 0.242·17-s − 0.229·19-s − 0.208·23-s + 0.928·29-s + 1.43·31-s + 0.493·37-s − 1.56·41-s − 0.304·43-s − 1.02·47-s − 3/7·49-s − 0.650·59-s + 1.02·61-s − 0.122·67-s + 1.42·71-s − 1.28·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-s − 1.16·89-s + 1.25·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3090411657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3090411657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.92051954659108, −12.43291334512991, −12.00034351421113, −11.60727553022120, −11.09703184825940, −10.22812521716827, −10.10821772635418, −9.840374375953649, −9.283590572674235, −8.563302480961046, −8.169902416399672, −7.789959319726358, −7.077292947951338, −6.750584837632058, −6.335676257923978, −5.623537970400586, −5.165544817834529, −4.638969884745952, −4.273749400280727, −3.395054410685790, −2.904933627196863, −2.587688911253166, −1.874604918404184, −1.093863830403981, −0.1593170562836757,
0.1593170562836757, 1.093863830403981, 1.874604918404184, 2.587688911253166, 2.904933627196863, 3.395054410685790, 4.273749400280727, 4.638969884745952, 5.165544817834529, 5.623537970400586, 6.335676257923978, 6.750584837632058, 7.077292947951338, 7.789959319726358, 8.169902416399672, 8.563302480961046, 9.283590572674235, 9.840374375953649, 10.10821772635418, 10.22812521716827, 11.09703184825940, 11.60727553022120, 12.00034351421113, 12.43291334512991, 12.92051954659108
