| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 5·11-s + 13-s + 15-s − 17-s − 6·19-s + 21-s − 6·23-s − 4·25-s − 27-s − 6·29-s − 4·31-s − 5·33-s + 35-s − 11·37-s − 39-s − 9·43-s − 45-s − 4·47-s + 49-s + 51-s + 7·53-s − 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.870·33-s + 0.169·35-s − 1.80·37-s − 0.160·39-s − 1.37·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.961·53-s − 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6226026624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6226026624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.53959863069009, −14.95957564896171, −14.50194257871636, −13.88453465933358, −13.18396543525394, −12.80641234429789, −11.99450482959751, −11.76440810465184, −11.29387774629768, −10.50633779499079, −10.11212926242094, −9.397799534714024, −8.802806414273613, −8.346493946357292, −7.537759283542464, −6.824193538196472, −6.511448852824902, −5.847898297368552, −5.248512250363538, −4.214348030775203, −3.937471760764010, −3.396477135443085, −2.035441823277326, −1.627234078018565, −0.3220730976635168,
0.3220730976635168, 1.627234078018565, 2.035441823277326, 3.396477135443085, 3.937471760764010, 4.214348030775203, 5.248512250363538, 5.847898297368552, 6.511448852824902, 6.824193538196472, 7.537759283542464, 8.346493946357292, 8.802806414273613, 9.397799534714024, 10.11212926242094, 10.50633779499079, 11.29387774629768, 11.76440810465184, 11.99450482959751, 12.80641234429789, 13.18396543525394, 13.88453465933358, 14.50194257871636, 14.95957564896171, 15.53959863069009
