| L(s) = 1 | + (−1.49 + 3.28i)2-s + (−3.28 − 3.78i)4-s + (−15.0 + 9.67i)5-s + (−2.56 + 17.8i)7-s + (−10.3 + 3.03i)8-s + (−9.18 − 63.8i)10-s + (10.3 + 22.7i)11-s + (7.78 + 54.1i)13-s + (−54.6 − 35.1i)14-s + (11.2 − 78.1i)16-s + (35.0 − 40.4i)17-s + (−11.3 − 13.0i)19-s + (86.0 + 25.2i)20-s − 90.1·22-s + (−26.7 − 107. i)23-s + ⋯ |
| L(s) = 1 | + (−0.529 + 1.16i)2-s + (−0.410 − 0.473i)4-s + (−1.34 + 0.864i)5-s + (−0.138 + 0.961i)7-s + (−0.457 + 0.134i)8-s + (−0.290 − 2.01i)10-s + (0.284 + 0.623i)11-s + (0.166 + 1.15i)13-s + (−1.04 − 0.670i)14-s + (0.175 − 1.22i)16-s + (0.500 − 0.577i)17-s + (−0.136 − 0.157i)19-s + (0.961 + 0.282i)20-s − 0.873·22-s + (−0.242 − 0.970i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.323204 - 0.245846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323204 - 0.245846i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (26.7 + 107. i)T \) |
| good | 2 | \( 1 + (1.49 - 3.28i)T + (-5.23 - 6.04i)T^{2} \) |
| 5 | \( 1 + (15.0 - 9.67i)T + (51.9 - 113. i)T^{2} \) |
| 7 | \( 1 + (2.56 - 17.8i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (-10.3 - 22.7i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-7.78 - 54.1i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (-35.0 + 40.4i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (11.3 + 13.0i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (55.4 - 64.0i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (100. - 29.3i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (160. + 102. i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-93.7 + 60.2i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-279. - 82.0i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (7.76 - 54.0i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-42.9 - 298. i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (800. - 235. i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (1.95 - 4.28i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (382. - 838. i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (165. + 190. i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-35.5 - 247. i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (828. + 532. i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (140. + 41.3i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-61.1 + 39.3i)T + (3.79e5 - 8.30e5i)T^{2} \) |
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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
−12.28073593870857660021135496683, −11.91114421123045963073770336148, −10.80797292326900253480243176524, −9.303538006927372086634637773341, −8.629518391145253106763246075880, −7.41739088010550840032044468007, −6.97310505020401181619363277462, −5.82096750631235558948747110851, −4.21045317777138156929506346771, −2.71483611002598654180312959479,
0.23771238804451677029296509544, 1.18667450542316819122244518168, 3.36734779520049069818686029172, 4.04237371921571921923778309434, 5.76357997659957974352970987613, 7.52159193650762771665456223299, 8.289260059293801744572202837775, 9.298272856724164106662521995420, 10.46910227390558912966299224307, 11.07841594685219141943544833124
