| L(s) = 1 | + 3-s − 3.90·5-s + 7-s + 9-s + 2.32·11-s + 1.20·13-s − 3.90·15-s + 4.70·17-s − 0.322·19-s + 21-s + 23-s + 10.2·25-s + 27-s − 5.10·29-s − 5.10·31-s + 2.32·33-s − 3.90·35-s − 2.70·37-s + 1.20·39-s + 7.48·41-s − 7.25·43-s − 3.90·45-s + 7.43·47-s + 49-s + 4.70·51-s + 3.28·53-s − 9.07·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.74·5-s + 0.377·7-s + 0.333·9-s + 0.700·11-s + 0.333·13-s − 1.00·15-s + 1.14·17-s − 0.0740·19-s + 0.218·21-s + 0.208·23-s + 2.05·25-s + 0.192·27-s − 0.948·29-s − 0.917·31-s + 0.404·33-s − 0.660·35-s − 0.444·37-s + 0.192·39-s + 1.16·41-s − 1.10·43-s − 0.582·45-s + 1.08·47-s + 0.142·49-s + 0.658·51-s + 0.451·53-s − 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943453241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943453241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.90T + 5T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 + 0.322T + 19T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 7.48T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 - 6.93T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 2.93T + 97T^{2} \) |
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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
−7.79001292442572374106684833060, −7.42146908710927231627093504943, −6.75777636037486287599058731018, −5.71392186914722013536533300855, −4.89023869941864676515081280436, −3.97298072572314170995093168672, −3.72494506399335820558394495349, −2.94190011370413016219703243069, −1.69932099711075962598421041903, −0.70025675861970126805346547275,
0.70025675861970126805346547275, 1.69932099711075962598421041903, 2.94190011370413016219703243069, 3.72494506399335820558394495349, 3.97298072572314170995093168672, 4.89023869941864676515081280436, 5.71392186914722013536533300855, 6.75777636037486287599058731018, 7.42146908710927231627093504943, 7.79001292442572374106684833060
