L(s) = 1 | − 3-s − 5-s + 4·7-s − 2·9-s + 6·11-s − 13-s + 15-s − 2·19-s − 4·21-s − 23-s + 25-s + 5·27-s + 9·29-s − 5·31-s − 6·33-s − 4·35-s + 2·37-s + 39-s − 9·41-s + 4·43-s + 2·45-s + 3·47-s + 9·49-s − 6·53-s − 6·55-s + 2·57-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s − 2/3·9-s + 1.80·11-s − 0.277·13-s + 0.258·15-s − 0.458·19-s − 0.872·21-s − 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s − 0.898·31-s − 1.04·33-s − 0.676·35-s + 0.328·37-s + 0.160·39-s − 1.40·41-s + 0.609·43-s + 0.298·45-s + 0.437·47-s + 9/7·49-s − 0.824·53-s − 0.809·55-s + 0.264·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.583303290\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.583303290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.992840943939303376021800431337, −8.533946654129671485375276039749, −7.74960017810923075341750161887, −6.77421173574083519058534893805, −6.10876762874484982420441460165, −5.04718544999915992036153081059, −4.47701693890377834352774704558, −3.50555919139989069203097099191, −2.05648279191482247682287196575, −0.925158674611848011972962077960,
0.925158674611848011972962077960, 2.05648279191482247682287196575, 3.50555919139989069203097099191, 4.47701693890377834352774704558, 5.04718544999915992036153081059, 6.10876762874484982420441460165, 6.77421173574083519058534893805, 7.74960017810923075341750161887, 8.533946654129671485375276039749, 8.992840943939303376021800431337