L(s) = 1 | + (−0.323 − 0.258i)2-s + (−0.403 − 0.0921i)3-s + (−0.406 − 1.78i)4-s + (0.623 − 0.781i)5-s + (0.106 + 0.134i)6-s + (−0.629 + 2.75i)7-s + (−0.688 + 1.42i)8-s + (−2.54 − 1.22i)9-s + (−0.403 + 0.0921i)10-s + (1.04 + 2.17i)11-s + 0.757i·12-s + (1.64 − 0.793i)13-s + (0.915 − 0.730i)14-s + (−0.323 + 0.258i)15-s + (−2.70 + 1.30i)16-s + 4.82i·17-s + ⋯ |
L(s) = 1 | + (−0.228 − 0.182i)2-s + (−0.233 − 0.0532i)3-s + (−0.203 − 0.891i)4-s + (0.278 − 0.349i)5-s + (0.0436 + 0.0547i)6-s + (−0.237 + 1.04i)7-s + (−0.243 + 0.505i)8-s + (−0.849 − 0.409i)9-s + (−0.127 + 0.0291i)10-s + (0.315 + 0.655i)11-s + 0.218i·12-s + (0.456 − 0.220i)13-s + (0.244 − 0.195i)14-s + (−0.0836 + 0.0666i)15-s + (−0.675 + 0.325i)16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.424320 + 0.376034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.424320 + 0.376034i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (0.323 + 0.258i)T + (0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.403 + 0.0921i)T + (2.70 + 1.30i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T + (-1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (0.629 - 2.75i)T + (-6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (-1.04 - 2.17i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.64 + 0.793i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 + (5.84 - 1.33i)T + (17.1 - 8.24i)T^{2} \) |
| 23 | \( 1 + (4.77 + 5.98i)T + (-5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-3.18 - 2.53i)T + (6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 + (1.73 - 3.60i)T + (-23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 - 12.4iT - 41T^{2} \) |
| 43 | \( 1 + (-5.01 + 3.99i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (2.27 + 4.72i)T + (-29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (4.66 - 5.85i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + (-0.807 - 0.184i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (5.09 + 2.45i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (2.85 - 1.37i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.12 - 2.49i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (0.179 - 0.373i)T + (-49.2 - 61.7i)T^{2} \) |
| 83 | \( 1 + (-0.813 - 3.56i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (3.50 + 2.79i)T + (19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (12.1 - 2.77i)T + (87.3 - 42.0i)T^{2} \) |
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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
−10.34447032966347146607445570273, −9.567595037303776873166483752456, −8.637050274947670634754800560944, −8.412796682229316455402754478218, −6.41287246854282769455556426251, −6.15447522453085905072462300313, −5.27789264817854425299602027833, −4.19147336955139490776690850372, −2.59747470227168368675803675473, −1.50703105554881294480179672699,
0.30577248043574266251307938777, 2.49572602629305816223201870729, 3.59933667168696382086877036159, 4.41278342854896205820777475067, 5.80011355313579690368944689259, 6.64501058870149498670123986838, 7.45068657042375342632070356427, 8.315669384957364452718779333465, 9.039322988527324652471043089894, 9.991216162986027289100854012628