L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 5·13-s − 14-s + 15-s + 16-s − 18-s − 20-s − 21-s + 2·22-s − 4·23-s + 24-s + 25-s − 5·26-s − 27-s + 28-s − 2·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8289711147\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8289711147\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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
−12.21808625873760, −12.00545452812050, −11.55088002973157, −10.95386648969001, −10.75537435427900, −10.43037181794465, −9.795391003866884, −9.327690822565850, −8.781010071497237, −8.382550532270109, −7.837428362119920, −7.708256545296235, −6.945168828262114, −6.526125809676472, −6.069378382043058, −5.562835916792654, −5.121378409687610, −4.425995460116119, −4.031209837024287, −3.322264015941654, −2.981955114910716, −1.983964534592848, −1.691075129422914, −0.9759171815616300, −0.3165686332861723,
0.3165686332861723, 0.9759171815616300, 1.691075129422914, 1.983964534592848, 2.981955114910716, 3.322264015941654, 4.031209837024287, 4.425995460116119, 5.121378409687610, 5.562835916792654, 6.069378382043058, 6.526125809676472, 6.945168828262114, 7.708256545296235, 7.837428362119920, 8.382550532270109, 8.781010071497237, 9.327690822565850, 9.795391003866884, 10.43037181794465, 10.75537435427900, 10.95386648969001, 11.55088002973157, 12.00545452812050, 12.21808625873760