| L(s) = 1 | − 3·3-s + 0.726·5-s + 9·9-s + 64.4·11-s + 71.8·13-s − 2.17·15-s + 48.9·17-s − 34.3·19-s + 0.903·23-s − 124.·25-s − 27·27-s + 226.·29-s + 275.·31-s − 193.·33-s + 295.·37-s − 215.·39-s + 186.·41-s + 455.·43-s + 6.53·45-s − 282.·47-s − 146.·51-s + 356.·53-s + 46.8·55-s + 103.·57-s − 729.·59-s + 274.·61-s + 52.1·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0649·5-s + 0.333·9-s + 1.76·11-s + 1.53·13-s − 0.0375·15-s + 0.697·17-s − 0.415·19-s + 0.00818·23-s − 0.995·25-s − 0.192·27-s + 1.45·29-s + 1.59·31-s − 1.02·33-s + 1.31·37-s − 0.884·39-s + 0.710·41-s + 1.61·43-s + 0.0216·45-s − 0.876·47-s − 0.402·51-s + 0.923·53-s + 0.114·55-s + 0.239·57-s − 1.60·59-s + 0.575·61-s + 0.0995·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.692574041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.692574041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.726T + 125T^{2} \) |
| 11 | \( 1 - 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.903T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 40.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 949.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 39.4T + 9.12e5T^{2} \) |
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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
−8.631758330810157468866240543703, −7.946506311135612143165786974882, −6.85926042035025068641917681205, −6.15913737499694432002054115953, −5.85506932340954660010160137992, −4.39206635207382981652551897374, −4.03849817760372907622874407004, −2.88358389202448346141835588957, −1.41508073595588284974086924793, −0.878803159018816621674037834486,
0.878803159018816621674037834486, 1.41508073595588284974086924793, 2.88358389202448346141835588957, 4.03849817760372907622874407004, 4.39206635207382981652551897374, 5.85506932340954660010160137992, 6.15913737499694432002054115953, 6.85926042035025068641917681205, 7.946506311135612143165786974882, 8.631758330810157468866240543703
