L(s) = 1 | + 2-s − 4-s + 3·5-s − 7-s − 3·8-s + 3·10-s − 7·13-s − 14-s − 16-s + 3·17-s + 2·19-s − 3·20-s + 4·23-s + 4·25-s − 7·26-s + 28-s + 7·29-s − 10·31-s + 5·32-s + 3·34-s − 3·35-s + 37-s + 2·38-s − 9·40-s − 5·41-s − 6·43-s + 4·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.34·5-s − 0.377·7-s − 1.06·8-s + 0.948·10-s − 1.94·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.670·20-s + 0.834·23-s + 4/5·25-s − 1.37·26-s + 0.188·28-s + 1.29·29-s − 1.79·31-s + 0.883·32-s + 0.514·34-s − 0.507·35-s + 0.164·37-s + 0.324·38-s − 1.42·40-s − 0.780·41-s − 0.914·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.27581301199820871973336952099, −6.77905483083942879019564812730, −5.84754319580687445290554192134, −5.32448512877908483824325215416, −4.96261310569353978083498661147, −4.01665610078146436223427920651, −2.99010340245957901390255403181, −2.56553042981486041986479700337, −1.39018804607000364893209284003, 0,
1.39018804607000364893209284003, 2.56553042981486041986479700337, 2.99010340245957901390255403181, 4.01665610078146436223427920651, 4.96261310569353978083498661147, 5.32448512877908483824325215416, 5.84754319580687445290554192134, 6.77905483083942879019564812730, 7.27581301199820871973336952099