| L(s) = 1 | − 1.41·5-s − 2.82·7-s − 4.24·13-s − 4·17-s + 8·19-s − 5.65·23-s − 2.99·25-s + 1.41·29-s + 2.82·31-s + 4.00·35-s − 4.24·37-s + 4·41-s − 11.3·47-s + 1.00·49-s − 7.07·53-s − 12·59-s − 1.41·61-s + 6·65-s − 4·67-s + 11.3·71-s + 14·73-s + 8.48·79-s − 16·83-s + 5.65·85-s − 6·89-s + 12·91-s − 11.3·95-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 1.06·7-s − 1.17·13-s − 0.970·17-s + 1.83·19-s − 1.17·23-s − 0.599·25-s + 0.262·29-s + 0.508·31-s + 0.676·35-s − 0.697·37-s + 0.624·41-s − 1.65·47-s + 0.142·49-s − 0.971·53-s − 1.56·59-s − 0.181·61-s + 0.744·65-s − 0.488·67-s + 1.34·71-s + 1.63·73-s + 0.954·79-s − 1.75·83-s + 0.613·85-s − 0.635·89-s + 1.25·91-s − 1.16·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6314553219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6314553219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−7.85051135596474282411277184443, −6.95863552327635429807377373134, −6.52555476265611447481759362629, −5.67147594439776898682701723090, −4.90135339268055628654269880337, −4.19616079737161093746243679368, −3.36271488682540483112672085126, −2.81381040486688729354367446694, −1.78058121932092624245849088188, −0.36513715477833542505483706902,
0.36513715477833542505483706902, 1.78058121932092624245849088188, 2.81381040486688729354367446694, 3.36271488682540483112672085126, 4.19616079737161093746243679368, 4.90135339268055628654269880337, 5.67147594439776898682701723090, 6.52555476265611447481759362629, 6.95863552327635429807377373134, 7.85051135596474282411277184443
