L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 3·11-s + 2·12-s + 13-s + 14-s + 16-s + 18-s − 19-s + 2·21-s + 3·22-s − 23-s + 2·24-s + 26-s − 4·27-s + 28-s − 3·29-s + 2·31-s + 32-s + 6·33-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.436·21-s + 0.639·22-s − 0.208·23-s + 0.408·24-s + 0.196·26-s − 0.769·27-s + 0.188·28-s − 0.557·29-s + 0.359·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.899104795\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.899104795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.634988328581323333596070153837, −8.899802507227352538447872909329, −8.184213334178221681134737084163, −7.37735148874630310057267571774, −6.43739865087035403286493821133, −5.50754155422734813561029289830, −4.32129922543763903254802790971, −3.63890914017928971738474067950, −2.64434246676066056155066635059, −1.59025346037222054836732618832,
1.59025346037222054836732618832, 2.64434246676066056155066635059, 3.63890914017928971738474067950, 4.32129922543763903254802790971, 5.50754155422734813561029289830, 6.43739865087035403286493821133, 7.37735148874630310057267571774, 8.184213334178221681134737084163, 8.899802507227352538447872909329, 9.634988328581323333596070153837