| L(s) = 1 | + (0.187 − 1.18i)2-s + (1.65 − 0.843i)3-s + (0.534 + 0.173i)4-s + (−2.13 + 0.666i)5-s + (−0.688 − 2.11i)6-s + (−3.01 − 3.01i)7-s + (1.39 − 2.73i)8-s + (0.264 − 0.364i)9-s + (0.389 + 2.65i)10-s + (3.77 − 2.74i)11-s + (1.03 − 0.163i)12-s + (0.156 + 0.987i)13-s + (−4.12 + 3.00i)14-s + (−2.97 + 2.90i)15-s + (−2.07 − 1.50i)16-s + (2.40 − 4.72i)17-s + ⋯ |
| L(s) = 1 | + (0.132 − 0.837i)2-s + (0.955 − 0.486i)3-s + (0.267 + 0.0868i)4-s + (−0.954 + 0.298i)5-s + (−0.280 − 0.864i)6-s + (−1.13 − 1.13i)7-s + (0.493 − 0.967i)8-s + (0.0882 − 0.121i)9-s + (0.123 + 0.838i)10-s + (1.13 − 0.826i)11-s + (0.297 − 0.0471i)12-s + (0.0433 + 0.273i)13-s + (−1.10 + 0.801i)14-s + (−0.766 + 0.749i)15-s + (−0.517 − 0.376i)16-s + (0.584 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.819329 - 1.62991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819329 - 1.62991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.13 - 0.666i)T \) |
| 19 | \( 1 + (3.96 + 1.80i)T \) |
| good | 2 | \( 1 + (-0.187 + 1.18i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-1.65 + 0.843i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (3.01 + 3.01i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.77 + 2.74i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.156 - 0.987i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 4.72i)T + (-9.99 - 13.7i)T^{2} \) |
| 23 | \( 1 + (0.503 - 3.17i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.74 - 8.43i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.28 + 2.04i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-9.94 + 1.57i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (4.13 - 5.68i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.07 + 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.49 - 1.27i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (8.76 - 4.46i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.92 - 4.30i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.69 - 2.68i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.30 - 2.19i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-9.47 - 3.07i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.656 + 4.14i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.59 - 8.00i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.58 + 2.33i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.54 + 1.12i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.84 - 7.54i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.98326616985089182312543902138, −9.861323299074592940238939822577, −9.007873872153108827504352255995, −7.82509884838685365384865589123, −7.10577224990634846314612528263, −6.49347273802817884165507416424, −4.21164984995171789311750043128, −3.42184201831688751652238665993, −2.82336920806049876786380368542, −1.01025174367057739178453449353,
2.31661999821999107191452474627, 3.55794236753949026296729748992, 4.48469244735897380935605673779, 6.06542556948690398579941166988, 6.51610168664434535976750628772, 7.948363453026498111006752929601, 8.406086417166483449982122505540, 9.345540031250747556022151817446, 10.11437399594712492717171676314, 11.49081519813999349090550879892
