L(s) = 1 | + 2-s − 3.32·3-s + 4-s + 5-s − 3.32·6-s + 2.40·7-s + 8-s + 8.08·9-s + 10-s − 5.27·11-s − 3.32·12-s + 6.65·13-s + 2.40·14-s − 3.32·15-s + 16-s + 5.66·17-s + 8.08·18-s + 7.64·19-s + 20-s − 8.01·21-s − 5.27·22-s − 2.99·23-s − 3.32·24-s + 25-s + 6.65·26-s − 16.9·27-s + 2.40·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.92·3-s + 0.5·4-s + 0.447·5-s − 1.35·6-s + 0.909·7-s + 0.353·8-s + 2.69·9-s + 0.316·10-s − 1.59·11-s − 0.960·12-s + 1.84·13-s + 0.643·14-s − 0.859·15-s + 0.250·16-s + 1.37·17-s + 1.90·18-s + 1.75·19-s + 0.223·20-s − 1.74·21-s − 1.12·22-s − 0.624·23-s − 0.679·24-s + 0.200·25-s + 1.30·26-s − 3.25·27-s + 0.454·28-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
\(L(1)\) |
\(\approx\) |
\(2.234884559\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.234884559\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.32T + 3T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 - 6.05T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 0.218T + 53T^{2} \) |
| 59 | \( 1 + 9.80T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 - 3.10T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 7.88T + 73T^{2} \) |
| 79 | \( 1 + 6.19T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−7.964982308951486501103467299224, −7.68304229820918141047310742699, −6.67108113145552504891849703597, −5.79704186457304718341953178692, −5.54863105683261201828244587271, −5.04997832855348650366089281922, −4.15975598273159876409189441907, −3.13661373163314290494192743348, −1.63677974015953010068298010429, −0.944199233329778596464483293508,
0.944199233329778596464483293508, 1.63677974015953010068298010429, 3.13661373163314290494192743348, 4.15975598273159876409189441907, 5.04997832855348650366089281922, 5.54863105683261201828244587271, 5.79704186457304718341953178692, 6.67108113145552504891849703597, 7.68304229820918141047310742699, 7.964982308951486501103467299224