L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 2·7-s − 8-s + 9-s + 4·10-s − 12-s + 2·14-s + 4·15-s + 16-s − 18-s − 4·20-s + 2·21-s + 6·23-s + 24-s + 11·25-s − 27-s − 2·28-s − 2·29-s − 4·30-s − 4·31-s − 32-s + 8·35-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.894·20-s + 0.436·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.730·30-s − 0.718·31-s − 0.176·32-s + 1.35·35-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.27207053902853, −12.81730120266640, −12.38446884589386, −12.04077960473897, −11.55090334701754, −11.01296320674949, −10.92170308489541, −10.29510311715427, −9.686584196533094, −9.236270089006208, −8.769087529163264, −8.138868931153656, −7.922839246673178, −7.146757830907490, −6.911377547991837, −6.640966617988389, −5.760747794173213, −5.253101793947774, −4.731510226607337, −3.950565346004397, −3.711110410555186, −3.046734897274113, −2.571941762009744, −1.503146483164414, −0.9478370296511583, 0, 0,
0.9478370296511583, 1.503146483164414, 2.571941762009744, 3.046734897274113, 3.711110410555186, 3.950565346004397, 4.731510226607337, 5.253101793947774, 5.760747794173213, 6.640966617988389, 6.911377547991837, 7.146757830907490, 7.922839246673178, 8.138868931153656, 8.769087529163264, 9.236270089006208, 9.686584196533094, 10.29510311715427, 10.92170308489541, 11.01296320674949, 11.55090334701754, 12.04077960473897, 12.38446884589386, 12.81730120266640, 13.27207053902853