L(s) = 1 | + 2-s + 1.74·3-s + 4-s − 5-s + 1.74·6-s + 3.19·7-s + 8-s + 0.0421·9-s − 10-s + 2.45·11-s + 1.74·12-s + 5.45·13-s + 3.19·14-s − 1.74·15-s + 16-s − 2.49·17-s + 0.0421·18-s + 2.96·19-s − 20-s + 5.58·21-s + 2.45·22-s − 1.55·23-s + 1.74·24-s + 25-s + 5.45·26-s − 5.15·27-s + 3.19·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.712·6-s + 1.20·7-s + 0.353·8-s + 0.0140·9-s − 0.316·10-s + 0.741·11-s + 0.503·12-s + 1.51·13-s + 0.855·14-s − 0.450·15-s + 0.250·16-s − 0.606·17-s + 0.00993·18-s + 0.680·19-s − 0.223·20-s + 1.21·21-s + 0.524·22-s − 0.324·23-s + 0.356·24-s + 0.200·25-s + 1.06·26-s − 0.992·27-s + 0.604·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.060787318\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.060787318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 1.74T + 3T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 6.95T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 + 6.84T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−8.301672610393036736919521708310, −7.86186661138549692685233713679, −7.10397132006299905946745266450, −6.10277857018206746552299456580, −5.47218894666315722811519873036, −4.32902419928444385625626106643, −3.93183633352688384785625689509, −3.12259974573650914514201740147, −2.10492131592638297921540065044, −1.24815080733874113189107892638,
1.24815080733874113189107892638, 2.10492131592638297921540065044, 3.12259974573650914514201740147, 3.93183633352688384785625689509, 4.32902419928444385625626106643, 5.47218894666315722811519873036, 6.10277857018206746552299456580, 7.10397132006299905946745266450, 7.86186661138549692685233713679, 8.301672610393036736919521708310