| L(s) = 1 | + (6.06 + 2.51i)3-s + (2.91 + 7.02i)5-s + (−13.3 + 13.3i)7-s + (11.3 + 11.3i)9-s + (49.6 − 20.5i)11-s + (−8.74 + 21.1i)13-s + 49.9i·15-s + 77.7i·17-s + (−53.3 + 128. i)19-s + (−114. + 47.5i)21-s + (−35.5 − 35.5i)23-s + (47.4 − 47.4i)25-s + (−27.3 − 66.0i)27-s + (245. + 101. i)29-s − 202.·31-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.483i)3-s + (0.260 + 0.628i)5-s + (−0.722 + 0.722i)7-s + (0.421 + 0.421i)9-s + (1.36 − 0.563i)11-s + (−0.186 + 0.450i)13-s + 0.859i·15-s + 1.10i·17-s + (−0.643 + 1.55i)19-s + (−1.19 + 0.494i)21-s + (−0.321 − 0.321i)23-s + (0.379 − 0.379i)25-s + (−0.195 − 0.470i)27-s + (1.57 + 0.650i)29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0479 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0479 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.522941168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.522941168\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-6.06 - 2.51i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (-2.91 - 7.02i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (13.3 - 13.3i)T - 343iT^{2} \) |
| 11 | \( 1 + (-49.6 + 20.5i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (8.74 - 21.1i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 - 77.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (53.3 - 128. i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (35.5 + 35.5i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-245. - 101. i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (36.3 + 87.6i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (36.8 + 36.8i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (185. - 76.8i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 - 82.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-534. + 221. i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (75.7 + 182. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-472. - 195. i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-102. - 42.5i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-520. + 520. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-244. - 244. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 774. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-23.9 + 57.7i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (351. - 351. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−11.94343029698826528457725847436, −10.57214878893305591848120668331, −9.801026096533839583358266096764, −8.856124592978095242634424008259, −8.341011320026771026813047613049, −6.66799292517516773493880339794, −6.00140239916683274048701495404, −4.02087196031513209721192136566, −3.26745498420062450836345087463, −2.00319630784530676268137146799,
0.889612144911993726927238811293, 2.41379282566852666002176909745, 3.66975211771488432517890820241, 4.93833608997283402212779762157, 6.69991285409613168275122197979, 7.27503408898165039897494361237, 8.602156533013310010298257974992, 9.231324397076031813478375353203, 10.03138782407098851986669010533, 11.46676206518629851374108414810
