| L(s) = 1 | + 3·3-s + 5·5-s − 12·7-s + 9·9-s − 37·11-s + 19·13-s + 15·15-s + 17·17-s − 37·19-s − 36·21-s + 3·23-s − 100·25-s + 27·27-s − 86·29-s + 142·31-s − 111·33-s − 60·35-s − 296·37-s + 57·39-s − 121·41-s − 3·43-s + 45·45-s − 402·47-s − 199·49-s + 51·51-s + 174·53-s − 185·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.647·7-s + 1/3·9-s − 1.01·11-s + 0.405·13-s + 0.258·15-s + 0.242·17-s − 0.446·19-s − 0.374·21-s + 0.0271·23-s − 4/5·25-s + 0.192·27-s − 0.550·29-s + 0.822·31-s − 0.585·33-s − 0.289·35-s − 1.31·37-s + 0.234·39-s − 0.460·41-s − 0.0106·43-s + 0.149·45-s − 1.24·47-s − 0.580·49-s + 0.140·51-s + 0.450·53-s − 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 17 | \( 1 - p T \) |
| good | 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 37 T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 19 | \( 1 + 37 T + p^{3} T^{2} \) |
| 23 | \( 1 - 3 T + p^{3} T^{2} \) |
| 29 | \( 1 + 86 T + p^{3} T^{2} \) |
| 31 | \( 1 - 142 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 121 T + p^{3} T^{2} \) |
| 43 | \( 1 + 3 T + p^{3} T^{2} \) |
| 47 | \( 1 + 402 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 270 T + p^{3} T^{2} \) |
| 61 | \( 1 + 520 T + p^{3} T^{2} \) |
| 67 | \( 1 - 780 T + p^{3} T^{2} \) |
| 71 | \( 1 + 84 T + p^{3} T^{2} \) |
| 73 | \( 1 + 302 T + p^{3} T^{2} \) |
| 79 | \( 1 + 178 T + p^{3} T^{2} \) |
| 83 | \( 1 + 698 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1512 T + p^{3} T^{2} \) |
| 97 | \( 1 + 500 T + p^{3} T^{2} \) |
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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
−9.527631166889175287803015797819, −8.546713322663855589593264972465, −7.84850381493504990610562111926, −6.81825601939711440099949022409, −5.94664781749912995542196372760, −4.96506671992925821990436689263, −3.70661741571984361827508374767, −2.80364861906883573424046295356, −1.70652967912283787220949947148, 0,
1.70652967912283787220949947148, 2.80364861906883573424046295356, 3.70661741571984361827508374767, 4.96506671992925821990436689263, 5.94664781749912995542196372760, 6.81825601939711440099949022409, 7.84850381493504990610562111926, 8.546713322663855589593264972465, 9.527631166889175287803015797819
