| L(s) = 1 | + 3-s − 2.34·5-s − 1.25·7-s + 9-s − 6.44·11-s + 1.34·13-s − 2.34·15-s − 7.39·17-s − 4.27·19-s − 1.25·21-s + 23-s + 0.499·25-s + 27-s − 29-s + 10.1·31-s − 6.44·33-s + 2.95·35-s − 5.20·37-s + 1.34·39-s − 10.6·41-s + 0.118·43-s − 2.34·45-s + 9.08·47-s − 5.41·49-s − 7.39·51-s + 10.4·53-s + 15.1·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.04·5-s − 0.475·7-s + 0.333·9-s − 1.94·11-s + 0.373·13-s − 0.605·15-s − 1.79·17-s − 0.981·19-s − 0.274·21-s + 0.208·23-s + 0.0998·25-s + 0.192·27-s − 0.185·29-s + 1.81·31-s − 1.12·33-s + 0.498·35-s − 0.856·37-s + 0.215·39-s − 1.65·41-s + 0.0180·43-s − 0.349·45-s + 1.32·47-s − 0.773·49-s − 1.03·51-s + 1.43·53-s + 2.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7081469475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7081469475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 17 | \( 1 + 7.39T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 0.118T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 - 7.85T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 7.91T + 83T^{2} \) |
| 89 | \( 1 + 4.52T + 89T^{2} \) |
| 97 | \( 1 + 0.279T + 97T^{2} \) |
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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
−7.984890516468071692518799725578, −7.16962430224819655163308201648, −6.68355369233497180668056127757, −5.77171701971693181523364907968, −4.77527350290997711756463512524, −4.31730943553791160291080596807, −3.44659722816928545714425508081, −2.71242766560783337534492626997, −2.04797189432331332621075817279, −0.37519141919393307256627573764,
0.37519141919393307256627573764, 2.04797189432331332621075817279, 2.71242766560783337534492626997, 3.44659722816928545714425508081, 4.31730943553791160291080596807, 4.77527350290997711756463512524, 5.77171701971693181523364907968, 6.68355369233497180668056127757, 7.16962430224819655163308201648, 7.984890516468071692518799725578
