L(s) = 1 | + (0.5 + 1.53i)2-s + (0.978 − 0.207i)3-s + (−0.5 + 0.363i)4-s + (0.190 − 0.330i)5-s + (0.809 + 1.40i)6-s + (2.74 + 1.22i)7-s + (1.80 + 1.31i)8-s + (−1.82 + 0.813i)9-s + (0.604 + 0.128i)10-s + (0.547 + 5.20i)11-s + (−0.413 + 0.459i)12-s + (−1.24 − 1.37i)13-s + (−0.507 + 4.82i)14-s + (0.118 − 0.363i)15-s + (−1.50 + 4.61i)16-s + (0.442 − 4.21i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 1.08i)2-s + (0.564 − 0.120i)3-s + (−0.250 + 0.181i)4-s + (0.0854 − 0.147i)5-s + (0.330 + 0.572i)6-s + (1.03 + 0.461i)7-s + (0.639 + 0.464i)8-s + (−0.609 + 0.271i)9-s + (0.191 + 0.0406i)10-s + (0.165 + 1.57i)11-s + (−0.119 + 0.132i)12-s + (−0.344 − 0.382i)13-s + (−0.135 + 1.29i)14-s + (0.0304 − 0.0937i)15-s + (−0.375 + 1.15i)16-s + (0.107 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77644 + 2.07337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77644 + 2.07337i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.978 + 0.207i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.190 + 0.330i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.74 - 1.22i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.547 - 5.20i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.24 + 1.37i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.442 + 4.21i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.34 + 3.71i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (2.80 + 2.04i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.97 + 6.06i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.11 - 3.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.41 - 0.513i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (1.59 - 1.77i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.73 - 5.34i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.646 + 0.288i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (0.516 - 0.109i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + (0.118 - 0.204i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (10.1 - 4.51i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (1.20 + 11.4i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (6.93 + 1.47i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-6.97 + 5.06i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-15.1 + 10.9i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.02210072621597689957686398344, −9.206286581503346255949415290352, −8.297292113873300619880767697851, −7.57731832057241107680884979617, −7.13441908420129117113620572991, −5.86595527908330031040092920227, −5.01937317403418002472148323647, −4.57506822597145748342164387213, −2.73136970837078241374868833766, −1.82131486956463051310435779230,
1.18893176681751076926950158809, 2.33158367423491637773942860434, 3.47383800564050540754393063253, 3.90991133505330854836096114206, 5.23514302420568479842552909160, 6.23603367592406476554416452818, 7.52009850035143660350602945880, 8.242323266400291895570192342123, 8.977344708401509629257355222115, 10.10178940174725705057293750257