L(s) = 1 | + 3.44·3-s + 9.69·5-s + 27.5·7-s − 15.1·9-s + 52.3·11-s − 5.40·13-s + 33.4·15-s + 17.1·17-s − 44.2·19-s + 95.1·21-s + 205.·23-s − 30.9·25-s − 145.·27-s + 29·29-s − 299.·31-s + 180.·33-s + 267.·35-s + 29.7·37-s − 18.6·39-s − 43.9·41-s − 64.8·43-s − 146.·45-s + 499.·47-s + 418.·49-s + 59.3·51-s − 351.·53-s + 507.·55-s + ⋯ |
L(s) = 1 | + 0.663·3-s + 0.867·5-s + 1.49·7-s − 0.559·9-s + 1.43·11-s − 0.115·13-s + 0.575·15-s + 0.245·17-s − 0.533·19-s + 0.989·21-s + 1.85·23-s − 0.247·25-s − 1.03·27-s + 0.185·29-s − 1.73·31-s + 0.952·33-s + 1.29·35-s + 0.132·37-s − 0.0765·39-s − 0.167·41-s − 0.229·43-s − 0.485·45-s + 1.55·47-s + 1.22·49-s + 0.162·51-s − 0.911·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.490308827\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.490308827\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
good | 3 | \( 1 - 3.44T + 27T^{2} \) |
| 5 | \( 1 - 9.69T + 125T^{2} \) |
| 7 | \( 1 - 27.5T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.40T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 205.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 29.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 351.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 504.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 481.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.11T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 295.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 428.T + 9.12e5T^{2} \) |
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
−10.79188765264643670074588831676, −9.361705426817524089501810848018, −8.980787959131185152213587447998, −8.058128312120464025977927412444, −7.01960784468949715470971004602, −5.84144626077110379918601811544, −4.91683029115285904396959059690, −3.66930167293221575639287909652, −2.26183433408648421913731492040, −1.33411931446612388888954597732,
1.33411931446612388888954597732, 2.26183433408648421913731492040, 3.66930167293221575639287909652, 4.91683029115285904396959059690, 5.84144626077110379918601811544, 7.01960784468949715470971004602, 8.058128312120464025977927412444, 8.980787959131185152213587447998, 9.361705426817524089501810848018, 10.79188765264643670074588831676