| L(s) = 1 | − 2i·3-s − 6i·5-s + 20·7-s + 23·9-s + 14i·11-s + 54i·13-s − 12·15-s − 66·17-s − 162i·19-s − 40i·21-s + 172·23-s + 89·25-s − 100i·27-s − 2i·29-s + 128·31-s + ⋯ |
| L(s) = 1 | − 0.384i·3-s − 0.536i·5-s + 1.07·7-s + 0.851·9-s + 0.383i·11-s + 1.15i·13-s − 0.206·15-s − 0.941·17-s − 1.95i·19-s − 0.415i·21-s + 1.55·23-s + 0.711·25-s − 0.712i·27-s − 0.0128i·29-s + 0.741·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.159927976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159927976\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2iT - 27T^{2} \) |
| 5 | \( 1 + 6iT - 125T^{2} \) |
| 7 | \( 1 - 20T + 343T^{2} \) |
| 11 | \( 1 - 14iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 54iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 162iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 172T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 + 158iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 202T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 408T + 1.03e5T^{2} \) |
| 53 | \( 1 - 690iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 322iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 298iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 202iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 700T + 3.57e5T^{2} \) |
| 73 | \( 1 - 418T + 3.89e5T^{2} \) |
| 79 | \( 1 + 744T + 4.93e5T^{2} \) |
| 83 | \( 1 - 678iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 82T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−11.44219998288083194948232067544, −10.73170998342869394310116125406, −9.226727554482896655655391608134, −8.720876015295427429896313739246, −7.30431033963732812057856582035, −6.75486291685059564076216632424, −4.90063092845972977905959074812, −4.45574923018987764525705688769, −2.28258213412362568623907717066, −1.05356037827133218037755388089,
1.35949382730584233736380939030, 3.07170465024721629943019246653, 4.39276264398433548185009259496, 5.42770628692440714392831437724, 6.76153876800283296971972619656, 7.83929309060385999874776174053, 8.679971080444354889778193315343, 10.08000811755738108822100566793, 10.64310340627630115152013957873, 11.48733560445079692264879132999
