L(s) = 1 | + 3-s − 2·4-s + 3·5-s − 7-s + 9-s − 2·12-s − 2·13-s + 3·15-s + 4·16-s − 3·17-s − 2·19-s − 6·20-s − 21-s − 23-s + 4·25-s + 27-s + 2·28-s + 8·31-s − 3·35-s − 2·36-s + 2·37-s − 2·39-s + 6·41-s − 5·43-s + 3·45-s + 9·47-s + 4·48-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 0.774·15-s + 16-s − 0.727·17-s − 0.458·19-s − 1.34·20-s − 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.377·28-s + 1.43·31-s − 0.507·35-s − 1/3·36-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.762·43-s + 0.447·45-s + 1.31·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58443 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58443 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.58379987114754, −13.95752544247532, −13.54290280573444, −13.19736464176036, −12.93345114274345, −12.11593978442951, −11.78135688064209, −10.69421399639032, −10.26833358486326, −9.990579611325523, −9.387742878761569, −8.948169631332744, −8.667838932000603, −7.915137369032592, −7.330952879501564, −6.661593810146156, −5.980185402322794, −5.721017640136938, −4.845418106926276, −4.418008509697495, −3.880550200977036, −2.875150634484994, −2.577287279367388, −1.771793171197445, −1.011287383874318, 0,
1.011287383874318, 1.771793171197445, 2.577287279367388, 2.875150634484994, 3.880550200977036, 4.418008509697495, 4.845418106926276, 5.721017640136938, 5.980185402322794, 6.661593810146156, 7.330952879501564, 7.915137369032592, 8.667838932000603, 8.948169631332744, 9.387742878761569, 9.990579611325523, 10.26833358486326, 10.69421399639032, 11.78135688064209, 12.11593978442951, 12.93345114274345, 13.19736464176036, 13.54290280573444, 13.95752544247532, 14.58379987114754