| L(s) = 1 | + 3·3-s + 13.6·5-s + 9·9-s + 32.8·11-s + 42.5·13-s + 40.8·15-s + 25.6·17-s − 15.2·19-s + 134.·23-s + 60.7·25-s + 27·27-s − 163.·29-s − 40.3·31-s + 98.6·33-s + 351.·37-s + 127.·39-s − 256.·41-s + 188·43-s + 122.·45-s − 290.·47-s + 76.8·51-s − 354.·53-s + 448.·55-s − 45.7·57-s + 616.·59-s − 394.·61-s + 579.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.21·5-s + 0.333·9-s + 0.901·11-s + 0.907·13-s + 0.703·15-s + 0.365·17-s − 0.184·19-s + 1.22·23-s + 0.486·25-s + 0.192·27-s − 1.04·29-s − 0.233·31-s + 0.520·33-s + 1.56·37-s + 0.523·39-s − 0.978·41-s + 0.666·43-s + 0.406·45-s − 0.900·47-s + 0.211·51-s − 0.919·53-s + 1.09·55-s − 0.106·57-s + 1.35·59-s − 0.828·61-s + 1.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.046584162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.046584162\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 13.6T + 125T^{2} \) |
| 11 | \( 1 - 32.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188T + 7.95e4T^{2} \) |
| 47 | \( 1 + 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 394.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 258.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 271.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
| 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
−9.351876026006502600000763520159, −8.808249297848529024273020357315, −7.83827267616385950168025450383, −6.77842162460289241381809068355, −6.11626315958103508904880424791, −5.23390278453547279884197897975, −4.04455475980947392188154331353, −3.10318335441387657518881507201, −1.95025978657252333264684548563, −1.11271841035295524852985346924,
1.11271841035295524852985346924, 1.95025978657252333264684548563, 3.10318335441387657518881507201, 4.04455475980947392188154331353, 5.23390278453547279884197897975, 6.11626315958103508904880424791, 6.77842162460289241381809068355, 7.83827267616385950168025450383, 8.808249297848529024273020357315, 9.351876026006502600000763520159
