L(s) = 1 | − 39.9i·2-s + 171. i·3-s − 1.08e3·4-s + 917.·5-s + 6.86e3·6-s + 835.·7-s + 2.28e4i·8-s − 9.83e3·9-s − 3.66e4i·10-s − 8.83e4i·11-s − 1.86e5i·12-s + 1.27e4·13-s − 3.33e4i·14-s + 1.57e5i·15-s + 3.58e5·16-s − 3.19e5i·17-s + ⋯ |
L(s) = 1 | − 1.76i·2-s + 1.22i·3-s − 2.11·4-s + 0.656·5-s + 2.16·6-s + 0.131·7-s + 1.97i·8-s − 0.499·9-s − 1.15i·10-s − 1.81i·11-s − 2.59i·12-s + 0.123·13-s − 0.232i·14-s + 0.804i·15-s + 1.36·16-s − 0.928i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.558239 - 1.51884i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.558239 - 1.51884i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-2.90e6 + 2.46e6i)T \) |
good | 2 | \( 1 + 39.9iT - 512T^{2} \) |
| 3 | \( 1 - 171. iT - 1.96e4T^{2} \) |
| 5 | \( 1 - 917.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 835.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.83e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.19e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 1.70e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 1.41e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 6.77e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + 7.64e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.65e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 7.74e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 3.23e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 7.70e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.10e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.38e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.80e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.52e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 2.14e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.66e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 - 1.26e9iT - 7.60e17T^{2} \) |
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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
−14.09061629348494825356568223297, −13.26281171933378757898860166528, −11.51994913950381558640318843741, −10.77744840468897589755551881024, −9.693208247943374351122456726199, −8.826867456328085137970090323269, −5.42076957356161969584020503588, −3.96120812255985781371509326369, −2.70722840681926081785240016611, −0.70829494399727991763714181135,
1.59668948116331836395848196032, 4.86238431664800607734956709111, 6.38969090563921563546324576554, 7.16225163589743094456094691844, 8.322823556950472802868484601587, 9.859994156497147794218412571370, 12.46969655702178075408329621904, 13.30843581446600530602110552692, 14.44622157103910110055499307512, 15.37982413950440360882538116593