L(s) = 1 | + (−3.54 + 8.56i)3-s + (−7.55 + 3.12i)5-s + (−7.16 + 7.16i)7-s + (−41.6 − 41.6i)9-s + (−0.758 − 1.83i)11-s + (71.0 + 29.4i)13-s − 75.7i·15-s − 98.5i·17-s + (−89.5 − 37.1i)19-s + (−35.9 − 86.7i)21-s + (−24.9 − 24.9i)23-s + (−41.1 + 41.1i)25-s + (273. − 113. i)27-s + (−57.8 + 139. i)29-s − 58.0·31-s + ⋯ |
L(s) = 1 | + (−0.682 + 1.64i)3-s + (−0.675 + 0.279i)5-s + (−0.386 + 0.386i)7-s + (−1.54 − 1.54i)9-s + (−0.0207 − 0.0501i)11-s + (1.51 + 0.628i)13-s − 1.30i·15-s − 1.40i·17-s + (−1.08 − 0.448i)19-s + (−0.373 − 0.901i)21-s + (−0.226 − 0.226i)23-s + (−0.329 + 0.329i)25-s + (1.95 − 0.808i)27-s + (−0.370 + 0.894i)29-s − 0.336·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.143938 - 0.253535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.143938 - 0.253535i\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (3.54 - 8.56i)T + (-19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (7.55 - 3.12i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (7.16 - 7.16i)T - 343iT^{2} \) |
| 11 | \( 1 + (0.758 + 1.83i)T + (-941. + 941. i)T^{2} \) |
| 13 | \( 1 + (-71.0 - 29.4i)T + (1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 + 98.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (89.5 + 37.1i)T + (4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (24.9 + 24.9i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (57.8 - 139. i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 + 58.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + (202. - 84.0i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (45.3 + 45.3i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (89.7 + 216. i)T + (-5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 - 4.38iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-8.98 - 21.6i)T + (-1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (287. - 119. i)T + (1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (28.2 - 68.0i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (293. - 708. i)T + (-2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-579. + 579. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (258. + 258. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 834. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (234. + 97.3i)T + (4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-179. + 179. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + 624.T + 9.12e5T^{2} \) |
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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
−13.67789588903524147618346886018, −12.10913833746893316716794342460, −11.23770412333349506515787360540, −10.67291680716501767803695972861, −9.397773140293870883974261874150, −8.676112647452904648540644803914, −6.76670800284071294609743583056, −5.56162138198210839969748968059, −4.31139230963495386695074091635, −3.32716371986607953148222005198,
0.16440523152190428156906800260, 1.63770384214424263103214222563, 3.83059231986260536631471455405, 5.84279383884856085925801538932, 6.53915379751039414352778333383, 7.87685757360368679541619326827, 8.414503709061757770740079313957, 10.49890678761307191803501646003, 11.34355548912746093267153136197, 12.41046431326553561764970818731