| L(s) = 1 | − 2.16·5-s + 2.64i·7-s + 4.55·13-s + 0.979i·19-s + 6i·23-s − 0.291·25-s − 5.74i·35-s − 7.00·49-s − 14.4i·59-s + 15.6·61-s − 9.87·65-s + 15.8i·71-s + 5.29i·79-s + 1.40i·83-s + 12.0i·91-s + ⋯ |
| L(s) = 1 | − 0.970·5-s + 0.999i·7-s + 1.26·13-s + 0.224i·19-s + 1.25i·23-s − 0.0583·25-s − 0.970i·35-s − 49-s − 1.87i·59-s + 1.99·61-s − 1.22·65-s + 1.88i·71-s + 0.595i·79-s + 0.153i·83-s + 1.26i·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9664676142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9664676142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 0.979iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.4iT - 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 1.40iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.450966751618832403959133514479, −8.218287893247582419282264791881, −7.33093609962854021810387109272, −6.49150528855246017558030064220, −5.72256462184676879468330636863, −5.08117652211847648725692838249, −3.90569339614366967409369649342, −3.51292924053768376423551777548, −2.38818028396425353795504477079, −1.25021409750481837125745185402,
0.31302888618116412492576568689, 1.33948921756011558131599940879, 2.74928804662955239816085269213, 3.80097903054943507630983816600, 4.09980904560530645618197530219, 5.02334062218648065070259765527, 6.11390405251822884081103115501, 6.77159061301848045741724166528, 7.48367741238172532086842062071, 8.163923918823487671544048894180
