L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s + 2·13-s − 14-s + 16-s + 18-s + 21-s + 22-s + 23-s − 24-s + 2·26-s − 27-s − 28-s − 6·29-s − 8·31-s + 32-s − 33-s + 36-s − 3·37-s − 2·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.493·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.94546876711597, −12.50256415207214, −12.00983663476746, −11.56290802362049, −11.13440631822783, −10.81552638193256, −10.26040104083912, −9.801086807322867, −9.236878335586016, −8.841459276210382, −8.244056788579502, −7.612246126156564, −7.196285799864097, −6.665947528485976, −6.378083554910528, −5.721964817145824, −5.324024149659517, −5.028672665109112, −4.162955059909950, −3.798785589926943, −3.457073075509556, −2.732575534160035, −1.998987388024904, −1.566872125006838, −0.7814752139623046, 0,
0.7814752139623046, 1.566872125006838, 1.998987388024904, 2.732575534160035, 3.457073075509556, 3.798785589926943, 4.162955059909950, 5.028672665109112, 5.324024149659517, 5.721964817145824, 6.378083554910528, 6.665947528485976, 7.196285799864097, 7.612246126156564, 8.244056788579502, 8.841459276210382, 9.236878335586016, 9.801086807322867, 10.26040104083912, 10.81552638193256, 11.13440631822783, 11.56290802362049, 12.00983663476746, 12.50256415207214, 12.94546876711597