L(s) = 1 | + (0.720 − 0.415i)2-s + (−1.53 − 2.65i)3-s + (−0.653 + 1.13i)4-s + 1.49i·5-s + (−2.20 − 1.27i)6-s + (1.75 + 1.01i)7-s + 2.75i·8-s + (−3.19 + 5.53i)9-s + (0.622 + 1.07i)10-s + (−4.84 + 2.79i)11-s + 4.00·12-s + (−3.59 + 0.284i)13-s + 1.68·14-s + (3.97 − 2.29i)15-s + (−0.163 − 0.282i)16-s + (0.509 − 0.881i)17-s + ⋯ |
L(s) = 1 | + (0.509 − 0.294i)2-s + (−0.884 − 1.53i)3-s + (−0.326 + 0.566i)4-s + 0.669i·5-s + (−0.901 − 0.520i)6-s + (0.661 + 0.381i)7-s + 0.972i·8-s + (−1.06 + 1.84i)9-s + (0.197 + 0.341i)10-s + (−1.46 + 0.843i)11-s + 1.15·12-s + (−0.996 + 0.0789i)13-s + 0.449·14-s + (1.02 − 0.592i)15-s + (−0.0407 − 0.0706i)16-s + (0.123 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.673732 + 0.419329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.673732 + 0.419329i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.59 - 0.284i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.720 + 0.415i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.53 + 2.65i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 1.49iT - 5T^{2} \) |
| 7 | \( 1 + (-1.75 - 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.84 - 2.79i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.509 + 0.881i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.64 - 3.25i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 2.34i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.95 + 3.38i)T + (-14.5 + 25.1i)T^{2} \) |
| 37 | \( 1 + (9.35 - 5.40i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.13 - 0.652i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.725 - 1.25i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.81iT - 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 + (-1.50 - 0.869i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.78 + 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.51 + 1.45i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.05 + 1.76i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 15.3iT - 83T^{2} \) |
| 89 | \( 1 + (-0.250 + 0.144i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.34 + 5.39i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.70131411449823642871601340955, −11.00529181271970910457054103221, −9.817333167300369029577589616879, −8.081833703946063823777053928166, −7.65754377357081898414646171163, −6.89389315248817425335562597867, −5.37483992938876141826258638270, −5.01984742917056200251267834688, −2.93531599165580247540233043095, −1.98739913078394706195657516194,
0.48199426253242342806675569036, 3.40391521592017943572212353539, 4.74743630430973405511943643086, 5.09391230354485416673118362392, 5.63151058930634449336382097140, 7.16801111573943381500010814667, 8.632096561314733496367353302739, 9.460239415494416988594893403174, 10.45536991790993298142558953021, 10.73863966062568131634640009674