L(s) = 1 | + 3-s − 1.41·7-s + 9-s + 1.41·13-s − 6·17-s − 6·19-s − 1.41·21-s + 8.48·23-s − 5·25-s + 27-s − 8.48·29-s + 1.41·31-s + 7.07·37-s + 1.41·39-s − 6·41-s − 6·43-s − 8.48·47-s − 5·49-s − 6·51-s + 8.48·53-s − 6·57-s − 7.07·61-s − 1.41·63-s + 4·67-s + 8.48·69-s − 8.48·71-s − 5·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.534·7-s + 0.333·9-s + 0.392·13-s − 1.45·17-s − 1.37·19-s − 0.308·21-s + 1.76·23-s − 25-s + 0.192·27-s − 1.57·29-s + 0.254·31-s + 1.16·37-s + 0.226·39-s − 0.937·41-s − 0.914·43-s − 1.23·47-s − 0.714·49-s − 0.840·51-s + 1.16·53-s − 0.794·57-s − 0.905·61-s − 0.178·63-s + 0.488·67-s + 1.02·69-s − 1.00·71-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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
−8.454853781367797756636553630687, −7.64626842078064387369624448832, −6.70630120667193617066883545007, −6.33653705252243361476186371196, −5.16788100506058689876652617387, −4.28502186995316375831689342527, −3.54164608461105587331387980920, −2.58580379467367364315928798204, −1.68386498468344546732835314810, 0,
1.68386498468344546732835314810, 2.58580379467367364315928798204, 3.54164608461105587331387980920, 4.28502186995316375831689342527, 5.16788100506058689876652617387, 6.33653705252243361476186371196, 6.70630120667193617066883545007, 7.64626842078064387369624448832, 8.454853781367797756636553630687