| L(s) = 1 | − 3·3-s − 5-s + 6·9-s − 2·11-s − 6·13-s + 3·15-s + 2·17-s − 9·23-s + 25-s − 9·27-s + 3·29-s + 2·31-s + 6·33-s + 8·37-s + 18·39-s + 5·41-s + 43-s − 6·45-s + 8·47-s − 6·51-s + 4·53-s + 2·55-s − 8·59-s + 7·61-s + 6·65-s − 3·67-s + 27·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s − 0.603·11-s − 1.66·13-s + 0.774·15-s + 0.485·17-s − 1.87·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s + 0.359·31-s + 1.04·33-s + 1.31·37-s + 2.88·39-s + 0.780·41-s + 0.152·43-s − 0.894·45-s + 1.16·47-s − 0.840·51-s + 0.549·53-s + 0.269·55-s − 1.04·59-s + 0.896·61-s + 0.744·65-s − 0.366·67-s + 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5248079969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5248079969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−10.15494162701926731661617056904, −9.570531777698586637991405174468, −7.980451556733360921500104168230, −7.44140215369026705115918097700, −6.44949735100563788876486177634, −5.64854776821868804577963207869, −4.87640735252252899437161046282, −4.10342107682854617221719588469, −2.40331970058823813257204091639, −0.59507167207658977792106133834,
0.59507167207658977792106133834, 2.40331970058823813257204091639, 4.10342107682854617221719588469, 4.87640735252252899437161046282, 5.64854776821868804577963207869, 6.44949735100563788876486177634, 7.44140215369026705115918097700, 7.980451556733360921500104168230, 9.570531777698586637991405174468, 10.15494162701926731661617056904
