| L(s) = 1 | − 2·11-s + 2·13-s + 6·17-s − 4·19-s + 6·23-s − 4·31-s − 10·37-s − 2·41-s − 4·43-s − 4·47-s + 12·53-s + 12·59-s − 6·61-s − 4·67-s + 14·71-s − 2·73-s + 8·79-s + 16·83-s + 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 0.718·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1.64·53-s + 1.56·59-s − 0.768·61-s − 0.488·67-s + 1.66·71-s − 0.234·73-s + 0.900·79-s + 1.75·83-s + 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.296690534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.296690534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.21341776481114, −12.72215617402318, −12.18641181551964, −11.95035601101209, −11.09647450302944, −10.90191459969243, −10.30752221545809, −10.02143683991549, −9.310214807956971, −8.886605516284232, −8.300393907158045, −8.053913770196463, −7.318082856286090, −6.908125367720827, −6.455646733055898, −5.701355821515643, −5.290943807712902, −4.984951783749791, −4.148901023048058, −3.519003154444597, −3.266798343208671, −2.451268111092639, −1.859937951928297, −1.164084790563806, −0.4644786895291451,
0.4644786895291451, 1.164084790563806, 1.859937951928297, 2.451268111092639, 3.266798343208671, 3.519003154444597, 4.148901023048058, 4.984951783749791, 5.290943807712902, 5.701355821515643, 6.455646733055898, 6.908125367720827, 7.318082856286090, 8.053913770196463, 8.300393907158045, 8.886605516284232, 9.310214807956971, 10.02143683991549, 10.30752221545809, 10.90191459969243, 11.09647450302944, 11.95035601101209, 12.18641181551964, 12.72215617402318, 13.21341776481114
