L(s) = 1 | + (−0.143 − 0.0595i)3-s + (0.767 + 1.85i)5-s + (−5.47 + 5.47i)7-s + (−19.0 − 19.0i)9-s + (36.9 − 15.2i)11-s + (−4.49 + 10.8i)13-s − 0.312i·15-s − 53.8i·17-s + (31.9 − 77.2i)19-s + (1.11 − 0.461i)21-s + (50.0 + 50.0i)23-s + (85.5 − 85.5i)25-s + (3.21 + 7.75i)27-s + (−156. − 64.6i)29-s + 207.·31-s + ⋯ |
L(s) = 1 | + (−0.0276 − 0.0114i)3-s + (0.0686 + 0.165i)5-s + (−0.295 + 0.295i)7-s + (−0.706 − 0.706i)9-s + (1.01 − 0.419i)11-s + (−0.0959 + 0.231i)13-s − 0.00537i·15-s − 0.768i·17-s + (0.386 − 0.932i)19-s + (0.0115 − 0.00479i)21-s + (0.454 + 0.454i)23-s + (0.684 − 0.684i)25-s + (0.0229 + 0.0553i)27-s + (−0.999 − 0.414i)29-s + 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.496565362\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496565362\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (0.143 + 0.0595i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (-0.767 - 1.85i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (5.47 - 5.47i)T - 343iT^{2} \) |
| 11 | \( 1 + (-36.9 + 15.2i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (4.49 - 10.8i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 53.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-31.9 + 77.2i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-50.0 - 50.0i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (156. + 64.6i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (73.7 + 177. i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (293. + 293. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-342. + 141. i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 + 510. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-590. + 244. i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-257. - 622. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (424. + 175. i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (482. + 199. i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (199. - 199. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (127. + 127. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 237. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (9.62 - 23.2i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-329. + 329. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 776.T + 9.12e5T^{2} \) |
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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
−11.71435332101210535092313664203, −10.47493022099932196654968046442, −9.142864445209445738837162491066, −8.901968005342709967673128626735, −7.24374736761356442107300943217, −6.39484674294759616701591310896, −5.33496316120683314808801308147, −3.80756642018060107982439015531, −2.63720910221607054034352787174, −0.63987216382560521020904171739,
1.41798778849386369765672491666, 3.10791748364977385635649255578, 4.43967533073259645780758700833, 5.66210621234348825598277260454, 6.73026613361062303682952910982, 7.890386959401593827722764413131, 8.840940896525426389153949570961, 9.877653041718949455929795142209, 10.78713671306678869910262020730, 11.74945003856756546286083510642