L(s) = 1 | + 9.17·3-s + 14.0·5-s + 13.6·7-s + 57.2·9-s + 4.50·11-s − 7.29·13-s + 128.·15-s + 28.0·17-s + 100.·19-s + 125.·21-s − 62.4·23-s + 71.2·25-s + 277.·27-s + 29·29-s − 156.·31-s + 41.3·33-s + 191.·35-s + 233.·37-s − 66.9·39-s + 7.39·41-s + 31.0·43-s + 801.·45-s − 226.·47-s − 156.·49-s + 257.·51-s − 254.·53-s + 63.1·55-s + ⋯ |
L(s) = 1 | + 1.76·3-s + 1.25·5-s + 0.737·7-s + 2.11·9-s + 0.123·11-s − 0.155·13-s + 2.21·15-s + 0.399·17-s + 1.20·19-s + 1.30·21-s − 0.566·23-s + 0.570·25-s + 1.97·27-s + 0.185·29-s − 0.909·31-s + 0.218·33-s + 0.924·35-s + 1.03·37-s − 0.274·39-s + 0.0281·41-s + 0.110·43-s + 2.65·45-s − 0.703·47-s − 0.456·49-s + 0.705·51-s − 0.658·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(6.880561806\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.880561806\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
good | 3 | \( 1 - 9.17T + 27T^{2} \) |
| 5 | \( 1 - 14.0T + 125T^{2} \) |
| 7 | \( 1 - 13.6T + 343T^{2} \) |
| 11 | \( 1 - 4.50T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.29T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.4T + 1.21e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 7.39T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 254.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 33.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 939.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 842.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 0.292T + 4.93e5T^{2} \) |
| 83 | \( 1 - 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 642.T + 9.12e5T^{2} \) |
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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
−9.029353822239741803094619135885, −8.006300905147196101340448091883, −7.69480295491587660102890373979, −6.63701958941021388454463887201, −5.60474408670308329768360310625, −4.72594245200637812771188703787, −3.65617295395862375634501029218, −2.79236278879885644449442708859, −1.95449338567705054928021922037, −1.30053867833231042195990594753,
1.30053867833231042195990594753, 1.95449338567705054928021922037, 2.79236278879885644449442708859, 3.65617295395862375634501029218, 4.72594245200637812771188703787, 5.60474408670308329768360310625, 6.63701958941021388454463887201, 7.69480295491587660102890373979, 8.006300905147196101340448091883, 9.029353822239741803094619135885