L(s) = 1 | + (14.0 + 24.3i)5-s + (75.7 − 131. i)7-s + (−138. + 240. i)11-s + (−291. − 505. i)13-s + 1.61e3·17-s + 1.36e3·19-s + (428. + 741. i)23-s + (1.16e3 − 2.02e3i)25-s + (4.26e3 − 7.39e3i)29-s + (−1.46e3 − 2.54e3i)31-s + 4.26e3·35-s + 4.03e3·37-s + (9.44e3 + 1.63e4i)41-s + (1.01e4 − 1.75e4i)43-s + (−147. + 256. i)47-s + ⋯ |
L(s) = 1 | + (0.251 + 0.435i)5-s + (0.583 − 1.01i)7-s + (−0.346 + 0.599i)11-s + (−0.479 − 0.829i)13-s + 1.35·17-s + 0.869·19-s + (0.168 + 0.292i)23-s + (0.373 − 0.646i)25-s + (0.942 − 1.63i)29-s + (−0.274 − 0.475i)31-s + 0.587·35-s + 0.484·37-s + (0.877 + 1.52i)41-s + (0.837 − 1.45i)43-s + (−0.00976 + 0.0169i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.96380 - 0.560988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96380 - 0.560988i\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + (-14.0 - 24.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-75.7 + 131. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (138. - 240. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (291. + 505. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.61e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-428. - 741. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-4.26e3 + 7.39e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.46e3 + 2.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 4.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.44e3 - 1.63e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.01e4 + 1.75e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (147. - 256. i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (8.61e3 + 1.49e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.28e4 - 2.22e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.31e4 - 2.27e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 7.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (4.96e4 - 8.59e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.50e4 + 4.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.33e4 + 5.77e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−12.69853523410872143270176217529, −11.57012826321025902377894069209, −10.35746189741087140115471632150, −9.819471289170027544304161568489, −7.941183645644945763073659514181, −7.33123996484941992271333666951, −5.74064842366782150389261634015, −4.40736139761940738563755020042, −2.78668236542198874491203066340, −0.925270396552731578992928963350,
1.33267262102304111631616495452, 2.96648729107531819423617426535, 4.91581690720870324513421030316, 5.75940827220070840408626649197, 7.40258500858005118752096314553, 8.638445038967365811634698061169, 9.436246216233107564930119884093, 10.82548846865451823614251572185, 11.95341126333950707879420601070, 12.64546200465706980083806451776