| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s − 3·14-s − 15-s + 16-s + 3·17-s + 18-s + 4·19-s + 20-s + 3·21-s − 4·22-s − 3·23-s − 24-s + 25-s + 26-s − 27-s − 3·28-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 − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.654·21-s − 0.852·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−12.72975887610249, −12.43846953309507, −12.09620727860669, −11.58515026946409, −11.03008576601198, −10.45253830050058, −10.26109753100532, −9.743775045685781, −9.444378515889261, −8.689745250892261, −8.091070295846446, −7.649956244869627, −7.153537047110735, −6.654002169669021, −6.118861141374719, −5.816791680526833, −5.387540756017697, −4.905673008369685, −4.347719971389949, −3.681002128071987, −3.231391365859695, −2.674688078562097, −2.300911415505053, −1.356629871074646, −0.8081393746901661, 0,
0.8081393746901661, 1.356629871074646, 2.300911415505053, 2.674688078562097, 3.231391365859695, 3.681002128071987, 4.347719971389949, 4.905673008369685, 5.387540756017697, 5.816791680526833, 6.118861141374719, 6.654002169669021, 7.153537047110735, 7.649956244869627, 8.091070295846446, 8.689745250892261, 9.444378515889261, 9.743775045685781, 10.26109753100532, 10.45253830050058, 11.03008576601198, 11.58515026946409, 12.09620727860669, 12.43846953309507, 12.72975887610249
