| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s − 2·13-s + 16-s − 6·17-s + 18-s − 4·22-s + 8·23-s − 24-s − 2·26-s − 27-s + 10·29-s + 8·31-s + 32-s + 4·33-s − 6·34-s + 36-s − 2·37-s + 2·39-s + 2·41-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.29527772193963476846365395682, −6.74553580824364922111766803824, −6.19274327191855656031230582072, −5.20857917839481535064914460677, −4.81913380955089112048934255178, −4.27105746059346910146373623514, −2.90278320711911551560660579265, −2.62440710274738908314805266739, −1.31092161404451362615182061580, 0,
1.31092161404451362615182061580, 2.62440710274738908314805266739, 2.90278320711911551560660579265, 4.27105746059346910146373623514, 4.81913380955089112048934255178, 5.20857917839481535064914460677, 6.19274327191855656031230582072, 6.74553580824364922111766803824, 7.29527772193963476846365395682
