L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 5-s + 4·6-s − 4·7-s + 9-s + 2·10-s − 6·11-s − 4·12-s − 4·13-s + 8·14-s + 2·15-s − 4·16-s − 6·17-s − 2·18-s − 6·19-s − 2·20-s + 8·21-s + 12·22-s − 23-s + 25-s + 8·26-s + 4·27-s − 8·28-s − 4·29-s − 4·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 0.447·5-s + 1.63·6-s − 1.51·7-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 1.15·12-s − 1.10·13-s + 2.13·14-s + 0.516·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.37·19-s − 0.447·20-s + 1.74·21-s + 2.55·22-s − 0.208·23-s + 1/5·25-s + 1.56·26-s + 0.769·27-s − 1.51·28-s − 0.742·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18745 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18745 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 163 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.70704900010773, −16.06760960952906, −15.96416849102095, −15.21148652232206, −14.73195286241110, −13.46888242283506, −13.04307100185556, −12.77419255756459, −12.02536459992862, −11.32758681885811, −10.86207765155416, −10.34415569528665, −10.08160172091473, −9.338818770141511, −8.706518613123044, −8.232487985150701, −7.346633491548248, −7.001816443631616, −6.468061664097225, −5.710934097476208, −4.996721311595689, −4.439573915625680, −3.353567864402950, −2.511759705456488, −1.858792131474545, 0, 0, 0,
1.858792131474545, 2.511759705456488, 3.353567864402950, 4.439573915625680, 4.996721311595689, 5.710934097476208, 6.468061664097225, 7.001816443631616, 7.346633491548248, 8.232487985150701, 8.706518613123044, 9.338818770141511, 10.08160172091473, 10.34415569528665, 10.86207765155416, 11.32758681885811, 12.02536459992862, 12.77419255756459, 13.04307100185556, 13.46888242283506, 14.73195286241110, 15.21148652232206, 15.96416849102095, 16.06760960952906, 16.70704900010773