| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 6·13-s − 15-s − 6·17-s − 6·19-s − 2·21-s − 2·23-s + 25-s − 27-s − 2·29-s + 4·31-s − 2·33-s + 2·35-s − 10·37-s + 6·39-s − 2·41-s + 8·43-s + 45-s + 6·47-s − 3·49-s + 6·51-s − 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.258·15-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.880832193208585765401663293455, −8.046181383612075719656590478555, −7.02737282613633366864692960599, −6.53199428258403666720874535161, −5.53548870346440664658998718897, −4.70355394163844512696991108806, −4.15763471966160038340013508239, −2.49145047879749491040660978061, −1.73508381507895261886045237038, 0,
1.73508381507895261886045237038, 2.49145047879749491040660978061, 4.15763471966160038340013508239, 4.70355394163844512696991108806, 5.53548870346440664658998718897, 6.53199428258403666720874535161, 7.02737282613633366864692960599, 8.046181383612075719656590478555, 8.880832193208585765401663293455
