| L(s) = 1 | + 1.73i·5-s − 3.46·7-s + 1.73i·13-s + 3·17-s − 2i·19-s − 6.92·23-s + 2.00·25-s − 8.66i·29-s − 3.46·31-s − 5.99i·35-s + 8.66i·37-s − 6·41-s − 4i·43-s + 3.46·47-s + 4.99·49-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s − 1.30·7-s + 0.480i·13-s + 0.727·17-s − 0.458i·19-s − 1.44·23-s + 0.400·25-s − 1.60i·29-s − 0.622·31-s − 1.01i·35-s + 1.42i·37-s − 0.937·41-s − 0.609i·43-s + 0.505·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026892298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026892298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 8.66iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 8.66iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 5.19iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.030490682930519164611479411124, −7.31724847569248112038824747298, −6.53566734918750165000942861191, −6.24233125827684609505635427679, −5.33217258365069066346072014921, −4.24677521253612454565948013894, −3.49878094511729205532547051818, −2.85527042338479343629720592059, −1.92839726827070503714512190686, −0.35504717286255685208212360787,
0.811170074160528653573777228183, 1.96029550852957378924245978869, 3.17616711571042104923054492612, 3.65590546536196443837190068297, 4.64647681968424460075705132195, 5.56172874512866653600924338355, 5.97676844471004406811245754097, 6.89904407818229285659357320835, 7.57588838751890288501364558037, 8.368473556461435432210937627210
