L(s) = 1 | + 3-s + 9-s − 13-s − 17-s − 6·19-s − 9·23-s + 27-s + 7·29-s + 7·31-s − 4·37-s − 39-s − 7·41-s + 43-s + 12·47-s − 51-s + 9·53-s − 6·57-s + 3·59-s − 9·61-s + 4·67-s − 9·69-s − 12·71-s + 8·73-s + 4·79-s + 81-s − 9·83-s + 7·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.242·17-s − 1.37·19-s − 1.87·23-s + 0.192·27-s + 1.29·29-s + 1.25·31-s − 0.657·37-s − 0.160·39-s − 1.09·41-s + 0.152·43-s + 1.75·47-s − 0.140·51-s + 1.23·53-s − 0.794·57-s + 0.390·59-s − 1.15·61-s + 0.488·67-s − 1.08·69-s − 1.42·71-s + 0.936·73-s + 0.450·79-s + 1/9·81-s − 0.987·83-s + 0.750·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.157440224\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157440224\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.27580501037923, −14.58428085281860, −14.01254436912012, −13.71836170432789, −13.16801859613761, −12.32305518907114, −12.15887199679326, −11.55960991876648, −10.60926089100795, −10.28365923109415, −9.924200890388075, −9.006231858160999, −8.628689455542750, −8.125112758045579, −7.582765013067211, −6.779858934817419, −6.378214431485824, −5.713326090873763, −4.864330671867758, −4.233974995996132, −3.842590971002202, −2.843535666556783, −2.330822687715843, −1.649881199524883, −0.5297171112072957,
0.5297171112072957, 1.649881199524883, 2.330822687715843, 2.843535666556783, 3.842590971002202, 4.233974995996132, 4.864330671867758, 5.713326090873763, 6.378214431485824, 6.779858934817419, 7.582765013067211, 8.125112758045579, 8.628689455542750, 9.006231858160999, 9.924200890388075, 10.28365923109415, 10.60926089100795, 11.55960991876648, 12.15887199679326, 12.32305518907114, 13.16801859613761, 13.71836170432789, 14.01254436912012, 14.58428085281860, 15.27580501037923