| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 7-s + 8-s + 9-s + 2·10-s + 2·11-s − 2·12-s − 4·13-s + 14-s − 4·15-s + 16-s + 6·17-s + 18-s + 2·20-s − 2·21-s + 2·22-s − 2·24-s − 25-s − 4·26-s + 4·27-s + 28-s − 2·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.447·20-s − 0.436·21-s + 0.426·22-s − 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s − 0.371·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.741084634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.741084634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.60821221820336577790892416467, −7.02555177218682477525777938215, −6.13473731467700445988773601277, −5.84388696943192329446894069998, −5.12768921784649198679752116202, −4.67926489077139327570246676481, −3.66298258523852573646031480507, −2.68936476020173603847106516371, −1.79608625711131815655643687833, −0.814795099828760560790460168235,
0.814795099828760560790460168235, 1.79608625711131815655643687833, 2.68936476020173603847106516371, 3.66298258523852573646031480507, 4.67926489077139327570246676481, 5.12768921784649198679752116202, 5.84388696943192329446894069998, 6.13473731467700445988773601277, 7.02555177218682477525777938215, 7.60821221820336577790892416467
