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
−11.49081519813999349090550879892, −10.11437399594712492717171676314, −9.345540031250747556022151817446, −8.406086417166483449982122505540, −7.948363453026498111006752929601, −6.51610168664434535976750628772, −6.06542556948690398579941166988, −4.48469244735897380935605673779, −3.55794236753949026296729748992, −2.31661999821999107191452474627,
1.01025174367057739178453449353, 2.82336920806049876786380368542, 3.42184201831688751652238665993, 4.21164984995171789311750043128, 6.49347273802817884165507416424, 7.10577224990634846314612528263, 7.82509884838685365384865589123, 9.007873872153108827504352255995, 9.861323299074592940238939822577, 10.98326616985089182312543902138