| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s − 4·23-s + 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 10·37-s + 2·39-s − 2·41-s + 4·43-s + 45-s − 4·47-s − 2·51-s + 10·53-s − 4·55-s − 4·59-s + 2·61-s − 2·65-s + 4·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s − 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.88976051029609, −15.08125548964841, −14.75778626210540, −14.07842744899901, −13.43678812063013, −13.05860162131316, −12.36085705776891, −12.12772001193315, −11.18589564159920, −10.89764111741930, −10.17702296478314, −9.819287670533553, −9.269401825328224, −8.418818632850347, −7.800701337761215, −7.389812797324052, −6.635088396197596, −5.949650372864857, −5.467118849243745, −4.974385229624458, −4.242780867475218, −3.442365924600698, −2.543422126517037, −2.036857419370655, −0.9511969090386866, 0,
0.9511969090386866, 2.036857419370655, 2.543422126517037, 3.442365924600698, 4.242780867475218, 4.974385229624458, 5.467118849243745, 5.949650372864857, 6.635088396197596, 7.389812797324052, 7.800701337761215, 8.418818632850347, 9.269401825328224, 9.819287670533553, 10.17702296478314, 10.89764111741930, 11.18589564159920, 12.12772001193315, 12.36085705776891, 13.05860162131316, 13.43678812063013, 14.07842744899901, 14.75778626210540, 15.08125548964841, 15.88976051029609
