L(s) = 1 | + 2.10·2-s + 2.44·4-s − 0.492·5-s − 7-s + 0.948·8-s − 1.03·10-s − 5.30·13-s − 2.10·14-s − 2.89·16-s + 3.03·17-s + 4.66·19-s − 1.20·20-s + 5.63·23-s − 4.75·25-s − 11.1·26-s − 2.44·28-s + 6.92·29-s − 1.26·31-s − 8.01·32-s + 6.40·34-s + 0.492·35-s + 10.8·37-s + 9.84·38-s − 0.466·40-s − 1.44·41-s − 2.88·43-s + 11.8·46-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 1.22·4-s − 0.220·5-s − 0.377·7-s + 0.335·8-s − 0.328·10-s − 1.47·13-s − 0.563·14-s − 0.724·16-s + 0.736·17-s + 1.07·19-s − 0.269·20-s + 1.17·23-s − 0.951·25-s − 2.19·26-s − 0.462·28-s + 1.28·29-s − 0.227·31-s − 1.41·32-s + 1.09·34-s + 0.0832·35-s + 1.78·37-s + 1.59·38-s − 0.0738·40-s − 0.225·41-s − 0.439·43-s + 1.75·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.981910559\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.981910559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 + 0.492T + 5T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 - 5.63T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 5.94T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.49965404720177581545968347308, −7.12903140081360777597379616722, −6.28996795171137483358055171815, −5.59232108374984126242430421447, −4.99593324480137207985317441593, −4.43615820471209460992348273138, −3.54695613514106419914666694958, −2.93938933597894874049106621435, −2.28072027274975955038040179870, −0.77727463015021198974542548569,
0.77727463015021198974542548569, 2.28072027274975955038040179870, 2.93938933597894874049106621435, 3.54695613514106419914666694958, 4.43615820471209460992348273138, 4.99593324480137207985317441593, 5.59232108374984126242430421447, 6.28996795171137483358055171815, 7.12903140081360777597379616722, 7.49965404720177581545968347308