| L(s) = 1 | − 2.70i·2-s + i·3-s − 5.32·4-s + 2.70·6-s − 0.470i·7-s + 8.99i·8-s − 9-s − 3.18·11-s − 5.32i·12-s + 0.563i·13-s − 1.27·14-s + 13.7·16-s + 1.70i·17-s + 2.70i·18-s − 3.74·19-s + ⋯ |
| L(s) = 1 | − 1.91i·2-s + 0.577i·3-s − 2.66·4-s + 1.10·6-s − 0.177i·7-s + 3.18i·8-s − 0.333·9-s − 0.959·11-s − 1.53i·12-s + 0.156i·13-s − 0.340·14-s + 3.42·16-s + 0.413i·17-s + 0.637i·18-s − 0.858·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119234401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119234401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.70iT - 2T^{2} \) |
| 7 | \( 1 + 0.470iT - 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 0.563iT - 13T^{2} \) |
| 17 | \( 1 - 1.70iT - 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 2.26iT - 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 1.02iT - 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 6.72iT - 43T^{2} \) |
| 47 | \( 1 - 4.43iT - 47T^{2} \) |
| 53 | \( 1 + 7.05iT - 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 0.126T + 61T^{2} \) |
| 67 | \( 1 - 2.79iT - 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 9.78iT - 73T^{2} \) |
| 79 | \( 1 - 4.75T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 6.20T + 89T^{2} \) |
| 97 | \( 1 + 8.45iT - 97T^{2} \) |
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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
−9.384347361487840104190313121346, −8.453237633482029913209405358619, −8.085645273909733472617790934098, −6.47188028752867345099420919894, −5.29281127613941927822881109047, −4.57739653919276624745569580049, −3.92630436190447261774779946415, −2.88785266468315473259752765134, −2.23300136913200365893436803489, −0.77243789899780720746414981371,
0.65058531952741792479626995901, 2.60492746598672277963673600137, 3.98513604252989161359989192164, 4.99171367951153536567411634937, 5.54812504205790751245977692407, 6.44383602802573873364623514723, 6.97126157347769240396266019400, 7.82398599252505670838205864053, 8.320421453113560939324233148130, 8.948375695091865470263843323128
