| L(s) = 1 | + (−0.254 + 1.19i)2-s + (−1.60 − 1.78i)3-s + (0.459 + 0.204i)4-s + 0.244i·5-s + (2.54 − 1.46i)6-s + (−0.0246 + 0.0554i)7-s + (−1.80 + 2.47i)8-s + (−0.289 + 2.75i)9-s + (−0.292 − 0.0620i)10-s + (0.706 − 0.0742i)11-s + (−0.373 − 1.14i)12-s + (3.31 + 1.42i)13-s + (−0.0600 − 0.0436i)14-s + (0.435 − 0.392i)15-s + (−1.83 − 2.03i)16-s + (0.244 − 2.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.179 + 0.846i)2-s + (−0.927 − 1.03i)3-s + (0.229 + 0.102i)4-s + 0.109i·5-s + (1.03 − 0.599i)6-s + (−0.00932 + 0.0209i)7-s + (−0.636 + 0.875i)8-s + (−0.0964 + 0.917i)9-s + (−0.0923 − 0.0196i)10-s + (0.213 − 0.0223i)11-s + (−0.107 − 0.331i)12-s + (0.918 + 0.396i)13-s + (−0.0160 − 0.0116i)14-s + (0.112 − 0.101i)15-s + (−0.458 − 0.509i)16-s + (0.0592 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.976379 + 0.441542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.976379 + 0.441542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-3.31 - 1.42i)T \) |
| 31 | \( 1 + (5.02 - 2.39i)T \) |
| good | 2 | \( 1 + (0.254 - 1.19i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.60 + 1.78i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 - 0.244iT - 5T^{2} \) |
| 7 | \( 1 + (0.0246 - 0.0554i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.706 + 0.0742i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (-0.244 + 2.32i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-5.00 - 4.50i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-5.12 + 2.28i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-4.00 - 0.851i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-5.20 - 3.00i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.88 - 8.84i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.79 + 5.32i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (2.18 + 0.708i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.70 + 6.32i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.52 + 11.8i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.70 + 5.60i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.67 - 0.385i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.320 + 0.440i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (5.61 + 4.08i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.70 - 1.20i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.83 + 0.297i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (1.78 - 4.01i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−11.47205041935740834460773768924, −10.83442084995901567943879851045, −9.326647904684368928391665974810, −8.295473497974450752187666986066, −7.36178385419163727436436871414, −6.68799112544972094769133258440, −6.03468596332571638023303782744, −5.06277565000122034256753075940, −3.10865681287502414583053706034, −1.32098788521248414673960915647,
1.02033520817552641625012805222, 2.96726447682299940198216311689, 4.06639716795808175688183482080, 5.27895191039032192107444716774, 6.12745209164734242676735773009, 7.27500717796424185689338746356, 8.915758654310743345571577245704, 9.575896821641137931342225301678, 10.58225957815678666704877140617, 10.99824494685101410458123610957
