L(s) = 1 | + 3-s + 7-s + 9-s + 3·11-s + 2·13-s + 5·19-s + 21-s − 23-s − 5·25-s + 27-s + 6·29-s − 10·31-s + 3·33-s + 2·37-s + 2·39-s + 3·41-s − 4·43-s + 3·47-s + 49-s + 3·53-s + 5·57-s + 9·59-s − 61-s + 63-s + 8·67-s − 69-s − 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.554·13-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s + 0.522·33-s + 0.328·37-s + 0.320·39-s + 0.468·41-s − 0.609·43-s + 0.437·47-s + 1/7·49-s + 0.412·53-s + 0.662·57-s + 1.17·59-s − 0.128·61-s + 0.125·63-s + 0.977·67-s − 0.120·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.523853638\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.523853638\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.189788947258605933935331869482, −8.443213990882921331064081677077, −7.69745705751471858834560872906, −6.95757425194463873125338080359, −6.02868546854759450384276676241, −5.14912030319413352672660977254, −4.04657412260723563729951674702, −3.44417974306597968440702646671, −2.19021788414494617415704484426, −1.13895919371716840319352283575,
1.13895919371716840319352283575, 2.19021788414494617415704484426, 3.44417974306597968440702646671, 4.04657412260723563729951674702, 5.14912030319413352672660977254, 6.02868546854759450384276676241, 6.95757425194463873125338080359, 7.69745705751471858834560872906, 8.443213990882921331064081677077, 9.189788947258605933935331869482