L(s) = 1 | + 544i·2-s − 5.90e4i·3-s + 1.80e6·4-s + 3.21e7·6-s + 1.27e9i·7-s + 2.12e9i·8-s − 3.48e9·9-s − 7.75e10·11-s − 1.06e11i·12-s + 4.34e11i·13-s − 6.95e11·14-s + 2.62e12·16-s + 1.28e13i·17-s − 1.89e12i·18-s + 2.86e13·19-s + ⋯ |
L(s) = 1 | + 0.375i·2-s − 0.577i·3-s + 0.858·4-s + 0.216·6-s + 1.70i·7-s + 0.698i·8-s − 0.333·9-s − 0.901·11-s − 0.495i·12-s + 0.873i·13-s − 0.642·14-s + 0.596·16-s + 1.54i·17-s − 0.125i·18-s + 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.419965240\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.419965240\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.90e4iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 544iT - 2.09e6T^{2} \) |
| 7 | \( 1 - 1.27e9iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 7.75e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.34e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 1.28e13iT - 6.90e25T^{2} \) |
| 19 | \( 1 - 2.86e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.24e14iT - 3.94e28T^{2} \) |
| 29 | \( 1 - 5.16e13T + 5.13e30T^{2} \) |
| 31 | \( 1 - 8.92e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.39e16iT - 8.55e32T^{2} \) |
| 41 | \( 1 - 5.81e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.61e17iT - 2.00e34T^{2} \) |
| 47 | \( 1 + 1.60e17iT - 1.30e35T^{2} \) |
| 53 | \( 1 + 2.29e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 5.15e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.25e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 5.40e18iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 1.10e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.77e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 + 6.31e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.45e20iT - 1.99e40T^{2} \) |
| 89 | \( 1 + 1.37e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 3.24e20iT - 5.27e41T^{2} \) |
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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
−11.38234636389406829678087864220, −10.03852225109877040080750869067, −8.541235757411729844553343037071, −7.998880043031489796676519571838, −6.56264906461255927839662847157, −6.01059813443408068690578454424, −4.93842359767822846108341535453, −2.95103798589061202117787510295, −2.30717519189310408124077141484, −1.36865520901347859720162580677,
0.42319798172835665039815170535, 1.10151404710942036482373667321, 2.71105110265892208653284275635, 3.38333037202406607495552751445, 4.59071811986835303198888610619, 5.75224585574992692262112503065, 7.31842652121896445789252854343, 7.63616960798431706065656080745, 9.582957060177314940491300022074, 10.36042220760514300015902062344