| L(s) = 1 | + 0.732·3-s + 0.267·5-s + 0.732·7-s − 2.46·9-s − 1.26·11-s + 2.46·13-s + 0.196·15-s − 3.46·17-s + 3.26·19-s + 0.535·21-s − 4.92·25-s − 4·27-s − 3·29-s − 3.46·31-s − 0.928·33-s + 0.196·35-s − 2·37-s + 1.80·39-s + 1.53·41-s + 3.46·43-s − 0.660·45-s − 6.19·47-s − 6.46·49-s − 2.53·51-s + 1.73·53-s − 0.339·55-s + 2.39·57-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.119·5-s + 0.276·7-s − 0.821·9-s − 0.382·11-s + 0.683·13-s + 0.0506·15-s − 0.840·17-s + 0.749·19-s + 0.116·21-s − 0.985·25-s − 0.769·27-s − 0.557·29-s − 0.622·31-s − 0.161·33-s + 0.0331·35-s − 0.328·37-s + 0.288·39-s + 0.239·41-s + 0.528·43-s − 0.0984·45-s − 0.903·47-s − 0.923·49-s − 0.355·51-s + 0.237·53-s − 0.0458·55-s + 0.316·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 8.12T + 97T^{2} \) |
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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
−8.003116418817888260882751767804, −7.52599502818464084075921562559, −6.51855751183387033855044434407, −5.76899838497998682010660551643, −5.16402202469052431808031735317, −4.11649697859958076627849419472, −3.34536919539902904999186797120, −2.47675449043392257928176633108, −1.56583096685929030785987702066, 0,
1.56583096685929030785987702066, 2.47675449043392257928176633108, 3.34536919539902904999186797120, 4.11649697859958076627849419472, 5.16402202469052431808031735317, 5.76899838497998682010660551643, 6.51855751183387033855044434407, 7.52599502818464084075921562559, 8.003116418817888260882751767804
