| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s + 2·11-s + 2·13-s − 15-s − 17-s − 6·19-s − 3·21-s + 23-s + 25-s + 27-s + 3·29-s + 3·31-s + 2·33-s + 3·35-s + 3·37-s + 2·39-s − 7·41-s − 45-s + 2·49-s − 51-s − 5·53-s − 2·55-s − 6·57-s − 9·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 0.538·31-s + 0.348·33-s + 0.507·35-s + 0.493·37-s + 0.320·39-s − 1.09·41-s − 0.149·45-s + 2/7·49-s − 0.140·51-s − 0.686·53-s − 0.269·55-s − 0.794·57-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.977021121404291806558389624607, −6.83029628755377452503535018869, −6.64098420640977711759245571463, −5.81967194850946087445855160896, −4.60962996579713013067034839989, −4.00009205857166505708746988493, −3.28158113120580597417931534021, −2.54354833888006110578393568912, −1.36173606385246130251257067448, 0,
1.36173606385246130251257067448, 2.54354833888006110578393568912, 3.28158113120580597417931534021, 4.00009205857166505708746988493, 4.60962996579713013067034839989, 5.81967194850946087445855160896, 6.64098420640977711759245571463, 6.83029628755377452503535018869, 7.977021121404291806558389624607
