| L(s) = 1 | + (1 − i)5-s + i·9-s + (−1 − i)13-s − i·25-s + (1 + i)29-s + (−1 + i)37-s + (1 + i)45-s − 49-s + (−1 + i)53-s + (−1 − i)61-s − 2·65-s − 2i·73-s − 81-s + 2i·89-s + (−1 + i)101-s + ⋯ |
| L(s) = 1 | + (1 − i)5-s + i·9-s + (−1 − i)13-s − i·25-s + (1 + i)29-s + (−1 + i)37-s + (1 + i)45-s − 49-s + (−1 + i)53-s + (−1 − i)61-s − 2·65-s − 2i·73-s − 81-s + 2i·89-s + (−1 + i)101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9889169800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9889169800\) |
| \(L(1)\) |
|
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 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.81756781478172153962242378919, −10.15043055657853386654155527562, −9.347566642105896422471237742116, −8.402508296750849753481275938973, −7.60882412645365566501785577679, −6.32597878781826611580469818195, −5.14070114452799738959413968177, −4.86880179487703982613096512255, −2.93003232488750337979992290562, −1.63950217480211767011410079303,
2.02620485764896993911860201981, 3.13092489716078216113276679951, 4.47279629603127473204875943311, 5.82278207144768728102714042620, 6.60586125186225878051442102548, 7.24890261465533674246434260554, 8.674244096584311672650023972001, 9.679783687589512220912685936009, 10.02323151848678624816619634562, 11.15590197353539919270439272000
