L(s) = 1 | − i·3-s + 0.331i·5-s − 3.08·7-s − 9-s − 3.69i·11-s + 4.64i·13-s + 0.331·15-s + 6.52·17-s + 0.867i·19-s + 3.08i·21-s − 4·23-s + 4.88·25-s + i·27-s − 4.89i·29-s + 6.14·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.148i·5-s − 1.16·7-s − 0.333·9-s − 1.11i·11-s + 1.28i·13-s + 0.0856·15-s + 1.58·17-s + 0.198i·19-s + 0.672i·21-s − 0.834·23-s + 0.977·25-s + 0.192i·27-s − 0.908i·29-s + 1.10·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.294580250\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.294580250\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 - 0.331iT - 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 3.69iT - 11T^{2} \) |
| 13 | \( 1 - 4.64iT - 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.867iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 - 3.64iT - 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 - 3.92iT - 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 0.564iT - 53T^{2} \) |
| 59 | \( 1 + 6.59iT - 59T^{2} \) |
| 61 | \( 1 + 14.8iT - 61T^{2} \) |
| 67 | \( 1 + 13.9iT - 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 + 5.62T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.35iT - 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.328353546833648328140397969813, −7.901745701272821423084551581363, −6.77596851898203877176026506190, −6.37662459664332065837522496877, −5.75727958290402909397713559138, −4.62618653835647402017082743563, −3.45226004029024139442581493255, −3.02207027965736448329444410041, −1.73051672577063483658067327866, −0.48602434227120361370740574953,
1.01098925357353978213060647551, 2.63276362975240387573131093836, 3.28787438005805520073519430184, 4.13336379356462313635475682379, 5.15769657412574718911156435797, 5.69274326584712132801298977760, 6.64365760106913406230793664869, 7.40607351535291022913153015192, 8.168101808693208602492118252424, 9.023860172245531862120306894784