L(s) = 1 | − 0.532·2-s − 1.71·4-s + 0.118·5-s − 7-s + 1.97·8-s − 0.0629·10-s + 5.22·11-s − 2.19·13-s + 0.532·14-s + 2.37·16-s + 1.28·17-s + 8.34·19-s − 0.202·20-s − 2.78·22-s + 9.05·23-s − 4.98·25-s + 1.16·26-s + 1.71·28-s + 9.26·29-s + 6.76·31-s − 5.22·32-s − 0.682·34-s − 0.118·35-s + 6.01·37-s − 4.44·38-s + 0.233·40-s + 4.32·41-s + ⋯ |
L(s) = 1 | − 0.376·2-s − 0.858·4-s + 0.0528·5-s − 0.377·7-s + 0.699·8-s − 0.0198·10-s + 1.57·11-s − 0.607·13-s + 0.142·14-s + 0.594·16-s + 0.310·17-s + 1.91·19-s − 0.0453·20-s − 0.593·22-s + 1.88·23-s − 0.997·25-s + 0.228·26-s + 0.324·28-s + 1.72·29-s + 1.21·31-s − 0.923·32-s − 0.117·34-s − 0.0199·35-s + 0.989·37-s − 0.720·38-s + 0.0369·40-s + 0.675·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.726304133\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.726304133\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 0.532T + 2T^{2} \) |
| 5 | \( 1 - 0.118T + 5T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 - 1.28T + 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 29 | \( 1 - 9.26T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 - 4.32T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 + 4.27T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 6.10T + 97T^{2} \) |
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
−7.77691059328800700879902220613, −7.32430607188947911998383569054, −6.46475382082623368698939597325, −5.81915523075257178397731282638, −4.80142958918882131220635365050, −4.49218653610256359809160474217, −3.42006399009620672957470394336, −2.88120147327784581202385663148, −1.32016569722918701813235200476, −0.829046382657367290042310738995,
0.829046382657367290042310738995, 1.32016569722918701813235200476, 2.88120147327784581202385663148, 3.42006399009620672957470394336, 4.49218653610256359809160474217, 4.80142958918882131220635365050, 5.81915523075257178397731282638, 6.46475382082623368698939597325, 7.32430607188947911998383569054, 7.77691059328800700879902220613