| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s − 6·11-s − 2·13-s − 15-s + 3·17-s − 6·19-s + 3·21-s − 23-s + 25-s + 27-s − 9·29-s − 3·31-s − 6·33-s − 3·35-s + 3·37-s − 2·39-s − 3·41-s − 45-s + 4·47-s + 2·49-s + 3·51-s − 9·53-s + 6·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.258·15-s + 0.727·17-s − 1.37·19-s + 0.654·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s − 0.538·31-s − 1.04·33-s − 0.507·35-s + 0.493·37-s − 0.320·39-s − 0.468·41-s − 0.149·45-s + 0.583·47-s + 2/7·49-s + 0.420·51-s − 1.23·53-s + 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.262094092030507477558648855943, −7.68684928921311552833726899639, −7.41171436382051265397325443603, −6.03708848837120204463217020448, −5.13725772231912507121855917288, −4.57700568815283346272628563695, −3.56715044059781580053321271420, −2.53954952861712960210514841900, −1.76815278521576512174257435821, 0,
1.76815278521576512174257435821, 2.53954952861712960210514841900, 3.56715044059781580053321271420, 4.57700568815283346272628563695, 5.13725772231912507121855917288, 6.03708848837120204463217020448, 7.41171436382051265397325443603, 7.68684928921311552833726899639, 8.262094092030507477558648855943
