| L(s) = 1 | + 3·3-s + 0.876·5-s − 7·7-s + 9·9-s − 56.3·11-s + 3.75·13-s + 2.63·15-s − 29.6·17-s + 135.·19-s − 21·21-s + 177.·23-s − 124.·25-s + 27·27-s + 149.·29-s + 38.4·31-s − 169.·33-s − 6.13·35-s − 86.2·37-s + 11.2·39-s + 58.5·41-s − 485.·43-s + 7.89·45-s − 315.·47-s + 49·49-s − 88.9·51-s − 154.·53-s − 49.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0784·5-s − 0.377·7-s + 0.333·9-s − 1.54·11-s + 0.0800·13-s + 0.0452·15-s − 0.422·17-s + 1.64·19-s − 0.218·21-s + 1.60·23-s − 0.993·25-s + 0.192·27-s + 0.960·29-s + 0.222·31-s − 0.891·33-s − 0.0296·35-s − 0.383·37-s + 0.0462·39-s + 0.222·41-s − 1.72·43-s + 0.0261·45-s − 0.979·47-s + 0.142·49-s − 0.244·51-s − 0.399·53-s − 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 0.876T + 125T^{2} \) |
| 11 | \( 1 + 56.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.75T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 58.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 485.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 315.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 154.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 727.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 688.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 710.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 777.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 853.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 231.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 444.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 634.T + 9.12e5T^{2} \) |
| 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
−8.857475343303291933438445380720, −7.964050528549049452103270508009, −7.38859423742354112666313899146, −6.44623458965923771446431970071, −5.34093169738954571921243240263, −4.68337856073924423988352858179, −3.24067909955114940558741024420, −2.81129250727026809476510284577, −1.44884726428428269713537835342, 0,
1.44884726428428269713537835342, 2.81129250727026809476510284577, 3.24067909955114940558741024420, 4.68337856073924423988352858179, 5.34093169738954571921243240263, 6.44623458965923771446431970071, 7.38859423742354112666313899146, 7.964050528549049452103270508009, 8.857475343303291933438445380720
