| L(s) = 1 | + (0.123 − 0.777i)2-s + (−2.41 + 1.22i)3-s + (1.31 + 0.426i)4-s + (−0.0275 + 2.23i)5-s + (0.658 + 2.02i)6-s + (−0.638 − 0.638i)7-s + (1.20 − 2.37i)8-s + (2.54 − 3.50i)9-s + (1.73 + 0.296i)10-s + (3.36 − 2.44i)11-s + (−3.68 + 0.584i)12-s + (0.663 + 4.18i)13-s + (−0.575 + 0.417i)14-s + (−2.68 − 5.42i)15-s + (0.537 + 0.390i)16-s + (−2.72 + 5.34i)17-s + ⋯ |
| L(s) = 1 | + (0.0871 − 0.549i)2-s + (−1.39 + 0.709i)3-s + (0.656 + 0.213i)4-s + (−0.0123 + 0.999i)5-s + (0.268 + 0.827i)6-s + (−0.241 − 0.241i)7-s + (0.427 − 0.838i)8-s + (0.847 − 1.16i)9-s + (0.548 + 0.0938i)10-s + (1.01 − 0.736i)11-s + (−1.06 + 0.168i)12-s + (0.184 + 1.16i)13-s + (−0.153 + 0.111i)14-s + (−0.692 − 1.40i)15-s + (0.134 + 0.0975i)16-s + (−0.660 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0854 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0854 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734081 + 0.673824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734081 + 0.673824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.0275 - 2.23i)T \) |
| 19 | \( 1 + (3.17 - 2.98i)T \) |
| good | 2 | \( 1 + (-0.123 + 0.777i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (2.41 - 1.22i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (0.638 + 0.638i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.36 + 2.44i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.663 - 4.18i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.72 - 5.34i)T + (-9.99 - 13.7i)T^{2} \) |
| 23 | \( 1 + (0.442 - 2.79i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.43 - 7.49i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.561 - 0.182i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.70 - 0.586i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.88 + 3.97i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (5.37 - 5.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.99 + 4.58i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.710 + 0.362i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.0 - 7.31i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.98 - 2.89i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.68 - 2.38i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-0.111 - 0.0363i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.12 + 7.09i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.53 + 4.71i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.10 - 1.07i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (14.7 - 10.7i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.00 + 1.97i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.16299998138970058871785409359, −10.63811836604210765410940043634, −9.997020624318292893285451329614, −8.744843259110705774650181660847, −7.06156677046525144688578597702, −6.49817157549601379907351054414, −5.83607364731333186142217227371, −4.05667383286560331818215598869, −3.64388394116692327771412489862, −1.76502335723699193344599920978,
0.70408923160363398280004236598, 2.17856077207713866950692848383, 4.52892196669979892450832823625, 5.36825479713290582875336464708, 6.16702506261983338552481656450, 6.87096584115095682641696813948, 7.71550755307951920785168529137, 8.899989337017401190012502642114, 10.02276299924350004267346742712, 11.15476709546851052217880294321
