| L(s) = 1 | + (3 − 3i)2-s + (−33 − 33i)3-s + 46i·4-s − 198·6-s + (−207 + 207i)7-s + (330 + 330i)8-s + 1.44e3i·9-s − 1.18e3·11-s + (1.51e3 − 1.51e3i)12-s + (−1.54e3 − 1.54e3i)13-s + 1.24e3i·14-s − 964·16-s + (−3.25e3 + 3.25e3i)17-s + (4.34e3 + 4.34e3i)18-s − 5.06e3i·19-s + ⋯ |
| L(s) = 1 | + (0.375 − 0.375i)2-s + (−1.22 − 1.22i)3-s + 0.718i·4-s − 0.916·6-s + (−0.603 + 0.603i)7-s + (0.644 + 0.644i)8-s + 1.98i·9-s − 0.892·11-s + (0.878 − 0.878i)12-s + (−0.704 − 0.704i)13-s + 0.452i·14-s − 0.235·16-s + (−0.661 + 0.661i)17-s + (0.745 + 0.745i)18-s − 0.737i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0775626 + 0.139116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0775626 + 0.139116i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-3 + 3i)T - 64iT^{2} \) |
| 3 | \( 1 + (33 + 33i)T + 729iT^{2} \) |
| 7 | \( 1 + (207 - 207i)T - 1.17e5iT^{2} \) |
| 11 | \( 1 + 1.18e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (1.54e3 + 1.54e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + (3.25e3 - 3.25e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + 5.06e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (5.31e3 + 5.31e3i)T + 1.48e8iT^{2} \) |
| 29 | \( 1 - 8.91e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.54e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (2.05e4 - 2.05e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 + 1.90e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (8.03e4 + 8.03e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + (1.61e4 - 1.61e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 + (-1.55e5 - 1.55e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + 3.60e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.78e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-2.40e5 + 2.40e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + 6.17e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.06e5 - 3.06e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 - 2.32e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-1.34e5 - 1.34e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + 2.70e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (8.10e5 - 8.10e5i)T - 8.32e11iT^{2} \) |
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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
−17.03374945810575154472730607488, −15.68644103252545880299581062755, −13.41379621889855893192789596838, −12.69409839766919909930606696825, −11.94532382312440808784018451552, −10.65593553826417411375623383060, −8.136683634775104805483940522097, −6.75593220014583264108774158144, −5.19042902279429821506950027959, −2.46434151181163290786222444525,
0.090292566058029499237041814117, 4.29086123585002579548855242742, 5.43961925487041518927652471242, 6.78124567014774622682948562712, 9.744926089166204726415768693034, 10.35987351510633542802621697143, 11.68386717168745406523330498739, 13.47921896995508316857243844212, 14.94699795722024897117586725429, 16.00020283524653599765689827358
