| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s + 2·17-s − 18-s − 19-s + 20-s − 21-s − 8·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + 2·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.218·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.63882281254709, −14.35741531073278, −13.75060935147535, −13.15630314629585, −12.48691870370207, −12.11534717691348, −11.62456366574555, −11.11163197849968, −10.40988470847465, −10.19289906316842, −9.686555928038025, −8.992256292182756, −8.554018818130177, −7.792248443504636, −7.547250059690548, −6.701817260676931, −6.340782769621022, −5.611370422783644, −5.278875270608034, −4.473278689172738, −3.823329385639759, −3.058892645831639, −2.115866749498765, −1.774418005606661, −0.8658101092352603, 0,
0.8658101092352603, 1.774418005606661, 2.115866749498765, 3.058892645831639, 3.823329385639759, 4.473278689172738, 5.278875270608034, 5.611370422783644, 6.340782769621022, 6.701817260676931, 7.547250059690548, 7.792248443504636, 8.554018818130177, 8.992256292182756, 9.686555928038025, 10.19289906316842, 10.40988470847465, 11.11163197849968, 11.62456366574555, 12.11534717691348, 12.48691870370207, 13.15630314629585, 13.75060935147535, 14.35741531073278, 14.63882281254709
