| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s − 12-s − 6·13-s − 3·14-s + 15-s + 16-s + 17-s + 18-s + 6·19-s − 20-s + 3·21-s + 3·23-s − 24-s + 25-s − 6·26-s − 27-s − 3·28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.66·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.654·21-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.566·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.67396729288826, −14.01027819669549, −13.28026440439382, −13.12280476497464, −12.34818544483524, −12.12021185607594, −11.71818048557736, −11.15333296326003, −10.43585255187904, −9.964917602659108, −9.655953985942755, −8.983028877697682, −8.241461733737537, −7.334973045996003, −7.251153596553235, −6.762785723686535, −6.006061960011147, −5.468270510429296, −4.975815066172715, −4.505156406981248, −3.656645564177027, −3.183483116481309, −2.692452526211269, −1.788747096049604, −0.7982203237088740, 0,
0.7982203237088740, 1.788747096049604, 2.692452526211269, 3.183483116481309, 3.656645564177027, 4.505156406981248, 4.975815066172715, 5.468270510429296, 6.006061960011147, 6.762785723686535, 7.251153596553235, 7.334973045996003, 8.241461733737537, 8.983028877697682, 9.655953985942755, 9.964917602659108, 10.43585255187904, 11.15333296326003, 11.71818048557736, 12.12021185607594, 12.34818544483524, 13.12280476497464, 13.28026440439382, 14.01027819669549, 14.67396729288826
