| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 3·14-s − 15-s + 16-s − 18-s + 5·19-s − 20-s − 3·21-s + 6·23-s − 24-s + 25-s + 26-s + 27-s − 3·28-s − 6·29-s + 30-s − 5·31-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 − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.654·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.566·28-s − 1.11·29-s + 0.182·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.81527395200954, −14.68950977940347, −13.63978944366943, −13.31048234369225, −12.82754534045620, −12.28652089392971, −11.69726609862627, −11.13765809322945, −10.70353476703074, −9.850319610532548, −9.605811619229486, −9.203221619343533, −8.635811082479163, −7.879220779954218, −7.593343291369794, −6.848050076314550, −6.660714732935374, −5.707262099703113, −5.180367095274312, −4.346758192574391, −3.461307905231652, −3.256066423541329, −2.570053742461046, −1.712892767717023, −0.8655826669678121, 0,
0.8655826669678121, 1.712892767717023, 2.570053742461046, 3.256066423541329, 3.461307905231652, 4.346758192574391, 5.180367095274312, 5.707262099703113, 6.660714732935374, 6.848050076314550, 7.593343291369794, 7.879220779954218, 8.635811082479163, 9.203221619343533, 9.605811619229486, 9.850319610532548, 10.70353476703074, 11.13765809322945, 11.69726609862627, 12.28652089392971, 12.82754534045620, 13.31048234369225, 13.63978944366943, 14.68950977940347, 14.81527395200954
