| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 4·11-s + 12-s − 2·14-s + 16-s + 18-s + 4·19-s − 2·21-s − 4·22-s + 4·23-s + 24-s + 27-s − 2·28-s − 6·29-s + 8·31-s + 32-s − 4·33-s + 36-s − 6·37-s + 4·38-s − 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.534·14-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.436·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s − 0.986·37-s + 0.648·38-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.99468748865383, −14.37364756284508, −13.70814847254199, −13.49568731701282, −13.01725447328973, −12.49121029405369, −12.02157803971423, −11.37029772778900, −10.72679176560113, −10.31297908389360, −9.660483376330399, −9.306937963176839, −8.490776754446548, −7.961868908878587, −7.465658288573403, −6.834613729370798, −6.397581858961742, −5.561599628904338, −5.105469695634212, −4.602378245999620, −3.596571200340139, −3.309078167381157, −2.703163329706208, −2.059096766482701, −1.119769776275490, 0,
1.119769776275490, 2.059096766482701, 2.703163329706208, 3.309078167381157, 3.596571200340139, 4.602378245999620, 5.105469695634212, 5.561599628904338, 6.397581858961742, 6.834613729370798, 7.465658288573403, 7.961868908878587, 8.490776754446548, 9.306937963176839, 9.660483376330399, 10.31297908389360, 10.72679176560113, 11.37029772778900, 12.02157803971423, 12.49121029405369, 13.01725447328973, 13.49568731701282, 13.70814847254199, 14.37364756284508, 14.99468748865383
