L(s) = 1 | + (−1.43e4 − 1.34e4i)3-s + 4.80e5i·5-s − 7.51e6·7-s + (2.31e7 + 3.86e8i)9-s + 2.15e9i·11-s − 1.26e9·13-s + (6.48e9 − 6.88e9i)15-s − 1.24e11i·17-s + 1.23e11·19-s + (1.07e11 + 1.01e11i)21-s − 1.00e11i·23-s + 3.58e12·25-s + (4.88e12 − 5.85e12i)27-s − 2.73e13i·29-s + 3.40e13·31-s + ⋯ |
L(s) = 1 | + (−0.727 − 0.685i)3-s + 0.246i·5-s − 0.186·7-s + (0.0597 + 0.998i)9-s + 0.915i·11-s − 0.119·13-s + (0.168 − 0.179i)15-s − 1.05i·17-s + 0.383·19-s + (0.135 + 0.127i)21-s − 0.0560i·23-s + 0.939·25-s + (0.640 − 0.767i)27-s − 1.88i·29-s + 1.28·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(1.337762062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.337762062\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.43e4 + 1.34e4i)T \) |
good | 5 | \( 1 - 4.80e5iT - 3.81e12T^{2} \) |
| 7 | \( 1 + 7.51e6T + 1.62e15T^{2} \) |
| 11 | \( 1 - 2.15e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 + 1.26e9T + 1.12e20T^{2} \) |
| 17 | \( 1 + 1.24e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 - 1.23e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + 1.00e11iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 2.73e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 - 3.40e13T + 6.99e26T^{2} \) |
| 37 | \( 1 - 1.11e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + 3.61e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 + 1.64e14T + 2.52e29T^{2} \) |
| 47 | \( 1 - 1.71e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 - 2.70e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + 6.75e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 - 7.85e15T + 1.36e32T^{2} \) |
| 67 | \( 1 - 3.25e16T + 7.40e32T^{2} \) |
| 71 | \( 1 + 4.47e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 7.24e16T + 3.46e33T^{2} \) |
| 79 | \( 1 - 2.22e16T + 1.43e34T^{2} \) |
| 83 | \( 1 - 2.78e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 + 6.88e15iT - 1.22e35T^{2} \) |
| 97 | \( 1 - 1.33e18T + 5.77e35T^{2} \) |
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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
−15.65296929383083420864467072529, −13.92807119471990712042709685207, −12.56343537334128393653812122666, −11.39697840328207400278757318616, −9.825761564912211119399744236161, −7.67335826336287834846462444409, −6.44470376444093978883204694362, −4.79393443911367223292649875916, −2.43938439935853465915442663729, −0.70097503350911959696629463528,
0.915192874751371619902568423125, 3.41459318496762096937772569327, 5.03494589168739918179038446549, 6.43173920076587753336401914773, 8.636474273217976319326825061078, 10.17519312385143122418085519265, 11.42231837269590145547744919172, 12.85896756278567840902066452724, 14.66807169825780722974819739186, 16.08244623959649606720306047247