| L(s) = 1 | + 1.84·5-s − 2i·11-s + 4.46i·13-s − 2.29·17-s + 1.53i·19-s − 8.82i·23-s − 1.58·25-s − 1.17i·29-s − 5.86i·31-s + 8.24·37-s − 11.8·41-s − 1.17·43-s − 8.02·47-s + 3.75i·53-s − 3.69i·55-s + ⋯ |
| L(s) = 1 | + 0.826·5-s − 0.603i·11-s + 1.23i·13-s − 0.556·17-s + 0.351i·19-s − 1.84i·23-s − 0.317·25-s − 0.217i·29-s − 1.05i·31-s + 1.35·37-s − 1.85·41-s − 0.178·43-s − 1.17·47-s + 0.516i·53-s − 0.498i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473275463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473275463\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 1.84T + 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 19 | \( 1 - 1.53iT - 19T^{2} \) |
| 23 | \( 1 + 8.82iT - 23T^{2} \) |
| 29 | \( 1 + 1.17iT - 29T^{2} \) |
| 31 | \( 1 + 5.86iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 + 8.02T + 47T^{2} \) |
| 53 | \( 1 - 3.75iT - 53T^{2} \) |
| 59 | \( 1 - 9.81T + 59T^{2} \) |
| 61 | \( 1 + 12.3iT - 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 + 2.74iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 2.74iT - 97T^{2} \) |
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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
−7.86151737473391766571204678492, −6.78759447787373616123268233824, −6.35952767943256227122443208725, −5.83238616689733227473115104403, −4.79310582886848733775469067581, −4.28615871383746932847493373677, −3.28965749163336580254259076168, −2.29093870557043999334601351219, −1.73183435099995322783380350634, −0.34085115419983131679941007933,
1.19650292133054703880367955097, 2.02652636946642069654432699520, 2.94129752733538838930472193053, 3.68531020541611421253443156388, 4.76406733191731137269189771476, 5.36009333019604719907376461136, 5.92991325376659934047612269162, 6.78404854958024259375359577268, 7.38197888936921316810669772466, 8.128314239574975591823038805100
