L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.35·7-s − 8-s + 9-s − 1.61·11-s − 12-s + 1.35·13-s + 3.35·14-s + 16-s − 6.96·17-s − 18-s − 19-s + 3.35·21-s + 1.61·22-s + 1.35·23-s + 24-s − 1.35·26-s − 27-s − 3.35·28-s + 3.61·29-s − 2.31·31-s − 32-s + 1.61·33-s + 6.96·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.26·7-s − 0.353·8-s + 0.333·9-s − 0.486·11-s − 0.288·12-s + 0.374·13-s + 0.895·14-s + 0.250·16-s − 1.68·17-s − 0.235·18-s − 0.229·19-s + 0.731·21-s + 0.343·22-s + 0.281·23-s + 0.204·24-s − 0.264·26-s − 0.192·27-s − 0.633·28-s + 0.670·29-s − 0.415·31-s − 0.176·32-s + 0.280·33-s + 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4959807754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4959807754\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 3.35T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.03T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.57T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
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
−8.934925939383128903158202441565, −8.129583550664170780109766942340, −7.04780761997897086621194786906, −6.65662885175712153736067539703, −5.96684596844018106182777058733, −5.01914658157131205870147332853, −3.96363138790030970759103552138, −2.98295707879308733740525477006, −1.97550494486033083755389630721, −0.47332244389407452799266575899,
0.47332244389407452799266575899, 1.97550494486033083755389630721, 2.98295707879308733740525477006, 3.96363138790030970759103552138, 5.01914658157131205870147332853, 5.96684596844018106182777058733, 6.65662885175712153736067539703, 7.04780761997897086621194786906, 8.129583550664170780109766942340, 8.934925939383128903158202441565