| L(s) = 1 | − i·5-s − i·9-s + (1 + i)13-s + (−1 + i)17-s − 25-s + (−1 + i)37-s − 45-s + i·49-s + (−1 − i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (1 + i)85-s + (1 − i)97-s + 2·101-s + ⋯ |
| L(s) = 1 | − i·5-s − i·9-s + (1 + i)13-s + (−1 + i)17-s − 25-s + (−1 + i)37-s − 45-s + i·49-s + (−1 − i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (1 + i)85-s + (1 − i)97-s + 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7909838631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7909838631\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−11.83342670478939600778000367090, −10.99823526752159599260567792653, −9.718829672758755078742899264270, −8.887964672811984972068635532586, −8.315241678892954349982762143012, −6.76267191062233945982580644459, −5.98658972706148741961646032438, −4.58466241709766521529310591437, −3.67342203931195572136873608633, −1.59942077213077209988873457183,
2.31263235827859086969458246725, 3.52007095924675564889294389541, 4.99971014154535995678329749748, 6.12915681482321798350687279061, 7.18830048997100790055883055519, 8.032732310269770728252854160536, 9.137370007529070708115527314309, 10.42385273045418601749797180161, 10.85084804999473395270961997714, 11.70730757930274210052070860578
