| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s − 6·11-s + 12-s + 2·13-s − 16-s − 4·17-s − 18-s + 6·19-s + 6·22-s − 3·24-s − 2·26-s − 27-s − 2·29-s + 10·31-s − 5·32-s + 6·33-s + 4·34-s − 36-s − 4·37-s − 6·38-s − 2·39-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.37·19-s + 1.27·22-s − 0.612·24-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.883·32-s + 1.04·33-s + 0.685·34-s − 1/6·36-s − 0.657·37-s − 0.973·38-s − 0.320·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.256055578461575445003684081516, −7.55446969326170854719470118234, −6.87365214021075367013688836973, −5.80251515017269393523381865598, −5.11647068828854224957302425430, −4.56813311176270756452064483969, −3.45367463617503835925378630300, −2.33462003232101750035985685627, −1.06504307972466057410251062188, 0,
1.06504307972466057410251062188, 2.33462003232101750035985685627, 3.45367463617503835925378630300, 4.56813311176270756452064483969, 5.11647068828854224957302425430, 5.80251515017269393523381865598, 6.87365214021075367013688836973, 7.55446969326170854719470118234, 8.256055578461575445003684081516
