| L(s) = 1 | − 4·11-s + 4·13-s − 4·19-s + 2·23-s − 2·29-s − 4·37-s + 2·41-s − 6·43-s − 6·47-s − 4·53-s + 12·59-s + 10·61-s + 14·67-s + 8·71-s + 8·73-s − 16·79-s + 2·83-s + 6·89-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1.10·13-s − 0.917·19-s + 0.417·23-s − 0.371·29-s − 0.657·37-s + 0.312·41-s − 0.914·43-s − 0.875·47-s − 0.549·53-s + 1.56·59-s + 1.28·61-s + 1.71·67-s + 0.949·71-s + 0.936·73-s − 1.80·79-s + 0.219·83-s + 0.635·89-s + 1.62·97-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
| \(L(1)\) |
\(\approx\) |
\(1.807630546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807630546\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.16801004031076, −12.84162613096048, −12.39831955698418, −11.60288402886246, −11.24475507869150, −10.90642682746033, −10.30780541459161, −9.990362731694000, −9.406338762356226, −8.720385419477743, −8.362945701543749, −8.079807349462837, −7.388277980078103, −6.795314285856960, −6.459865958016384, −5.778575322720139, −5.309207418738634, −4.879299266337675, −4.207373208736180, −3.548158257708209, −3.243049379634721, −2.308083489329652, −2.037537749600193, −1.126049050704487, −0.4110704697508722,
0.4110704697508722, 1.126049050704487, 2.037537749600193, 2.308083489329652, 3.243049379634721, 3.548158257708209, 4.207373208736180, 4.879299266337675, 5.309207418738634, 5.778575322720139, 6.459865958016384, 6.795314285856960, 7.388277980078103, 8.079807349462837, 8.362945701543749, 8.720385419477743, 9.406338762356226, 9.990362731694000, 10.30780541459161, 10.90642682746033, 11.24475507869150, 11.60288402886246, 12.39831955698418, 12.84162613096048, 13.16801004031076
