L(s) = 1 | + 16.7·3-s + 94.1·7-s + 36.4·9-s − 143.·11-s − 421.·13-s − 1.98e3·17-s + 1.31e3·19-s + 1.57e3·21-s − 4.02e3·23-s − 3.45e3·27-s − 6.41e3·29-s + 2.35e3·31-s − 2.40e3·33-s + 7.87e3·37-s − 7.04e3·39-s + 1.50e4·41-s + 1.14e3·43-s − 2.15e4·47-s − 7.94e3·49-s − 3.31e4·51-s − 9.56e3·53-s + 2.20e4·57-s − 4.27e4·59-s + 3.21e4·61-s + 3.42e3·63-s − 3.03e4·67-s − 6.71e4·69-s + ⋯ |
L(s) = 1 | + 1.07·3-s + 0.726·7-s + 0.149·9-s − 0.358·11-s − 0.691·13-s − 1.66·17-s + 0.837·19-s + 0.778·21-s − 1.58·23-s − 0.911·27-s − 1.41·29-s + 0.439·31-s − 0.383·33-s + 0.945·37-s − 0.741·39-s + 1.40·41-s + 0.0941·43-s − 1.42·47-s − 0.472·49-s − 1.78·51-s − 0.467·53-s + 0.897·57-s − 1.59·59-s + 1.10·61-s + 0.108·63-s − 0.826·67-s − 1.69·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 16.7T + 243T^{2} \) |
| 7 | \( 1 - 94.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.74e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.78e5T + 8.58e9T^{2} \) |
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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
−9.727780318277424483882787351702, −9.105337353307699089399334489608, −8.016284745952880963481572173553, −7.62900989047835736359077136896, −6.21838542525810187159018167494, −4.96302903621047775984481241829, −3.94350993734851807979424323114, −2.64186017399520297856882663150, −1.85660239731371396509878399809, 0,
1.85660239731371396509878399809, 2.64186017399520297856882663150, 3.94350993734851807979424323114, 4.96302903621047775984481241829, 6.21838542525810187159018167494, 7.62900989047835736359077136896, 8.016284745952880963481572173553, 9.105337353307699089399334489608, 9.727780318277424483882787351702