L(s) = 1 | − 1.99·2-s − 1.80·3-s + 2.98·4-s + 0.809·5-s + 3.59·6-s + 0.319·7-s − 3.95·8-s + 2.24·9-s − 1.61·10-s − 5.37·12-s + 1.43·13-s − 0.637·14-s − 1.45·15-s + 4.91·16-s + 1.60·17-s − 4.48·18-s + 1.03·19-s + 2.41·20-s − 0.575·21-s + 7.13·24-s − 0.344·25-s − 2.86·26-s − 2.24·27-s + 0.952·28-s + 2.91·30-s − 0.192·31-s − 5.85·32-s + ⋯ |
L(s) = 1 | − 1.99·2-s − 1.80·3-s + 2.98·4-s + 0.809·5-s + 3.59·6-s + 0.319·7-s − 3.95·8-s + 2.24·9-s − 1.61·10-s − 5.37·12-s + 1.43·13-s − 0.637·14-s − 1.45·15-s + 4.91·16-s + 1.60·17-s − 4.48·18-s + 1.03·19-s + 2.41·20-s − 0.575·21-s + 7.13·24-s − 0.344·25-s − 2.86·26-s − 2.24·27-s + 0.952·28-s + 2.91·30-s − 0.192·31-s − 5.85·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3625487558\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3625487558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1511 | \( 1 - T \) |
good | 2 | \( 1 + 1.99T + T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 - 0.809T + T^{2} \) |
| 7 | \( 1 - 0.319T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.43T + T^{2} \) |
| 17 | \( 1 - 1.60T + T^{2} \) |
| 19 | \( 1 - 1.03T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 0.192T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.96T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.74T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.796148218759566073839833099807, −9.148366317482131964019469678596, −7.996486861382337042142821257387, −7.38637312166726383887283856273, −6.37537555385114565435703382340, −5.94704541345175731638995535210, −5.30926715976886982053825278230, −3.37279203312191594271699371604, −1.65597387433535807567790158715, −1.03787905993172701895276717316,
1.03787905993172701895276717316, 1.65597387433535807567790158715, 3.37279203312191594271699371604, 5.30926715976886982053825278230, 5.94704541345175731638995535210, 6.37537555385114565435703382340, 7.38637312166726383887283856273, 7.996486861382337042142821257387, 9.148366317482131964019469678596, 9.796148218759566073839833099807