L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·11-s + 3·13-s + 14-s + 16-s − 7·17-s − 6·19-s − 4·22-s − 9·23-s + 3·26-s + 28-s − 3·29-s − 7·31-s + 32-s − 7·34-s + 10·37-s − 6·38-s + 41-s − 13·43-s − 4·44-s − 9·46-s + 2·47-s + 49-s + 3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 1.37·19-s − 0.852·22-s − 1.87·23-s + 0.588·26-s + 0.188·28-s − 0.557·29-s − 1.25·31-s + 0.176·32-s − 1.20·34-s + 1.64·37-s − 0.973·38-s + 0.156·41-s − 1.98·43-s − 0.603·44-s − 1.32·46-s + 0.291·47-s + 1/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−8.290251213252795354974819961741, −7.55882959742601779602174367794, −6.60483939538498540399741279532, −6.00319196493276518524450406425, −5.21976972007899643601957955048, −4.31251430197882907994876452276, −3.79422720845184208699867130096, −2.44484456080504610576362563646, −1.91741028052992770612375915587, 0,
1.91741028052992770612375915587, 2.44484456080504610576362563646, 3.79422720845184208699867130096, 4.31251430197882907994876452276, 5.21976972007899643601957955048, 6.00319196493276518524450406425, 6.60483939538498540399741279532, 7.55882959742601779602174367794, 8.290251213252795354974819961741