| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 4·11-s − 12-s − 6·13-s − 2·14-s + 15-s + 16-s + 18-s + 2·19-s − 20-s + 2·21-s − 4·22-s − 23-s − 24-s − 4·25-s − 6·26-s − 27-s − 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 642 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 642 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 107 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−10.13075362161529521985442973840, −9.675939185100960136955050953414, −8.045232506749785170052767772894, −7.35941913278579244983454854491, −6.46198133746891100484101184249, −5.38508550055242831936429997317, −4.73601267781378194181480807875, −3.47275145751272543922949372461, −2.36899373377871972661831920342, 0,
2.36899373377871972661831920342, 3.47275145751272543922949372461, 4.73601267781378194181480807875, 5.38508550055242831936429997317, 6.46198133746891100484101184249, 7.35941913278579244983454854491, 8.045232506749785170052767772894, 9.675939185100960136955050953414, 10.13075362161529521985442973840
