L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s − 2·11-s − 13-s − 2·15-s + 6·17-s − 6·19-s − 2·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s − 6·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s − 10·41-s − 6·43-s − 45-s + 49-s + 12·51-s + 6·53-s + 2·55-s − 12·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.516·15-s + 1.45·17-s − 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 1.07·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 1.56·41-s − 0.914·43-s − 0.149·45-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 0.269·55-s − 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.227193802250047829154823491701, −7.54235451365253913293399324500, −6.99386462287558887073401831828, −5.87438142998007025620890932184, −5.12359611515010124502172264638, −4.06330443400042223632693917042, −3.31523423051443757368793104229, −2.73567056252507547089907374018, −1.66517241218203840264688616008, 0,
1.66517241218203840264688616008, 2.73567056252507547089907374018, 3.31523423051443757368793104229, 4.06330443400042223632693917042, 5.12359611515010124502172264638, 5.87438142998007025620890932184, 6.99386462287558887073401831828, 7.54235451365253913293399324500, 8.227193802250047829154823491701