L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s + 7·13-s + 16-s − 7·17-s − 18-s − 8·19-s + 2·22-s + 5·23-s + 24-s − 7·26-s − 27-s + 9·29-s − 31-s − 32-s + 2·33-s + 7·34-s + 36-s + 2·37-s + 8·38-s − 7·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.94·13-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 1.83·19-s + 0.426·22-s + 1.04·23-s + 0.204·24-s − 1.37·26-s − 0.192·27-s + 1.67·29-s − 0.179·31-s − 0.176·32-s + 0.348·33-s + 1.20·34-s + 1/6·36-s + 0.328·37-s + 1.29·38-s − 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9046049823\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9046049823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.161579214378401291568829537170, −7.05843057825339532955080582849, −6.39458841310839291064788874771, −6.25382419699589388917998813277, −5.08266305835947659122195561684, −4.41715891375845601038983478015, −3.53251220016350761259141843826, −2.48945010567147348115376006582, −1.62787447261425645296710471064, −0.55616979618488413983403850452,
0.55616979618488413983403850452, 1.62787447261425645296710471064, 2.48945010567147348115376006582, 3.53251220016350761259141843826, 4.41715891375845601038983478015, 5.08266305835947659122195561684, 6.25382419699589388917998813277, 6.39458841310839291064788874771, 7.05843057825339532955080582849, 8.161579214378401291568829537170