L(s) = 1 | + 22i·5-s − 92i·13-s + 104·17-s − 359·25-s − 130i·29-s − 396i·37-s − 472·41-s − 343·49-s − 518i·53-s − 468i·61-s + 2.02e3·65-s + 1.09e3·73-s + 2.28e3i·85-s − 176·89-s + 594·97-s + ⋯ |
L(s) = 1 | + 1.96i·5-s − 1.96i·13-s + 1.48·17-s − 2.87·25-s − 0.832i·29-s − 1.75i·37-s − 1.79·41-s − 49-s − 1.34i·53-s − 0.982i·61-s + 3.86·65-s + 1.76·73-s + 2.91i·85-s − 0.209·89-s + 0.621·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.496798537\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496798537\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 22iT - 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 92iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 104T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 130iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 396iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 472T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 518iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 + 468iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 176T + 7.04e5T^{2} \) |
| 97 | \( 1 - 594T + 9.12e5T^{2} \) |
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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
−9.732953960746515101223919517471, −8.123635703419419196953461610066, −7.71420663486240630148559518068, −6.86029747532526183396515388647, −5.98153055837876300370539954559, −5.29994290890271469750969986721, −3.55210932264918301656539141648, −3.21580376133176961370314166557, −2.14246599443990054505614360189, −0.38698956432979135456994068457,
1.11774815787704034665666049057, 1.75864231969248505801397874591, 3.50755159231955498694552543596, 4.54563533469717558309558860576, 5.04436048771782344601069793970, 6.03161652659051841864626223642, 7.08276952125274223408447709788, 8.153446041973150164978631847715, 8.694770171897172745619544128182, 9.459575567983006801211560838401