L(s) = 1 | − 2-s − 2.59·3-s + 4-s − 0.859·5-s + 2.59·6-s − 3.29·7-s − 8-s + 3.72·9-s + 0.859·10-s − 0.214·11-s − 2.59·12-s + 0.577·13-s + 3.29·14-s + 2.22·15-s + 16-s + 6.01·17-s − 3.72·18-s + 0.272·19-s − 0.859·20-s + 8.54·21-s + 0.214·22-s + 2.84·23-s + 2.59·24-s − 4.26·25-s − 0.577·26-s − 1.88·27-s − 3.29·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.49·3-s + 0.5·4-s − 0.384·5-s + 1.05·6-s − 1.24·7-s − 0.353·8-s + 1.24·9-s + 0.271·10-s − 0.0646·11-s − 0.748·12-s + 0.160·13-s + 0.880·14-s + 0.575·15-s + 0.250·16-s + 1.45·17-s − 0.878·18-s + 0.0624·19-s − 0.192·20-s + 1.86·21-s + 0.0457·22-s + 0.592·23-s + 0.529·24-s − 0.852·25-s − 0.113·26-s − 0.363·27-s − 0.622·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5242889581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5242889581\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 0.859T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 13 | \( 1 - 0.577T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 - 0.272T + 19T^{2} \) |
| 23 | \( 1 - 2.84T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 + 0.438T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 3.37T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 + 0.683T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−7.80610487119845133485696466136, −6.86618844128654184672590916887, −6.62214076469832431517351941390, −5.79148710433968218709666949865, −5.37004770392155836105711525504, −4.34726947278900970834959717336, −3.45529188660064884997837095399, −2.69616056807863146645319961580, −1.21323140534805997614673559008, −0.50079929770644138123710232522,
0.50079929770644138123710232522, 1.21323140534805997614673559008, 2.69616056807863146645319961580, 3.45529188660064884997837095399, 4.34726947278900970834959717336, 5.37004770392155836105711525504, 5.79148710433968218709666949865, 6.62214076469832431517351941390, 6.86618844128654184672590916887, 7.80610487119845133485696466136