| L(s) = 1 | − 1.30·3-s + 2.30·5-s − 7-s − 1.30·9-s − 6.60·13-s − 3·15-s + 17-s + 6.60·19-s + 1.30·21-s + 0.302·25-s + 5.60·27-s − 4.30·31-s − 2.30·35-s + 2.60·37-s + 8.60·39-s + 3.90·41-s + 7.30·43-s − 3.00·45-s − 4.60·47-s + 49-s − 1.30·51-s + 3.69·53-s − 8.60·57-s + 9.21·59-s + 7.90·61-s + 1.30·63-s − 15.2·65-s + ⋯ |
| L(s) = 1 | − 0.752·3-s + 1.02·5-s − 0.377·7-s − 0.434·9-s − 1.83·13-s − 0.774·15-s + 0.242·17-s + 1.51·19-s + 0.284·21-s + 0.0605·25-s + 1.07·27-s − 0.772·31-s − 0.389·35-s + 0.428·37-s + 1.37·39-s + 0.610·41-s + 1.11·43-s − 0.447·45-s − 0.671·47-s + 0.142·49-s − 0.182·51-s + 0.507·53-s − 1.13·57-s + 1.19·59-s + 1.01·61-s + 0.164·63-s − 1.88·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−7.33174666925136630362952194753, −6.83922307129045932198163991185, −5.93297516164669373477741881550, −5.44162446925718712248612380019, −5.08475776723502891076083148394, −4.01242198527031390065436107263, −2.85562560309784182177288766388, −2.39167351789335839828403264526, −1.15898815137792494663967075663, 0,
1.15898815137792494663967075663, 2.39167351789335839828403264526, 2.85562560309784182177288766388, 4.01242198527031390065436107263, 5.08475776723502891076083148394, 5.44162446925718712248612380019, 5.93297516164669373477741881550, 6.83922307129045932198163991185, 7.33174666925136630362952194753
