| L(s) = 1 | − 2-s + 3-s + 4-s + 2.69·5-s − 6-s − 8-s + 9-s − 2.69·10-s − 4.49·11-s + 12-s − 13-s + 2.69·15-s + 16-s + 7.88·17-s − 18-s + 6.03·19-s + 2.69·20-s + 4.49·22-s + 2.65·23-s − 24-s + 2.24·25-s + 26-s + 27-s − 1.64·29-s − 2.69·30-s + 1.84·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.20·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.851·10-s − 1.35·11-s + 0.288·12-s − 0.277·13-s + 0.695·15-s + 0.250·16-s + 1.91·17-s − 0.235·18-s + 1.38·19-s + 0.602·20-s + 0.959·22-s + 0.552·23-s − 0.204·24-s + 0.449·25-s + 0.196·26-s + 0.192·27-s − 0.304·29-s − 0.491·30-s + 0.332·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215706782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215706782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 17 | \( 1 - 7.88T + 17T^{2} \) |
| 19 | \( 1 - 6.03T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 5.44T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 1.34T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 5.89T + 79T^{2} \) |
| 83 | \( 1 - 3.72T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 5.28T + 97T^{2} \) |
| 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
−8.490280637480951145294467647195, −7.70449154842625762907129939094, −7.40663111514819900707310241209, −6.31012649159763321530724993633, −5.42516718865908196106867367747, −5.10596891207913812161670712853, −3.43974694350590386277963479290, −2.82661965207183215195514900203, −1.95956489689644695261424235380, −0.968037822150481950309525785926,
0.968037822150481950309525785926, 1.95956489689644695261424235380, 2.82661965207183215195514900203, 3.43974694350590386277963479290, 5.10596891207913812161670712853, 5.42516718865908196106867367747, 6.31012649159763321530724993633, 7.40663111514819900707310241209, 7.70449154842625762907129939094, 8.490280637480951145294467647195
