| L(s) = 1 | − 2.06·3-s − 28.8·7-s − 22.7·9-s − 18.7·11-s − 86.7·13-s − 64.7·17-s − 27.2·19-s + 59.6·21-s + 102.·23-s + 102.·27-s + 8.87·29-s + 272.·31-s + 38.8·33-s − 82.4·37-s + 179.·39-s − 249.·41-s − 137.·43-s − 439.·47-s + 490.·49-s + 133.·51-s + 490.·53-s + 56.3·57-s + 530.·59-s − 407.·61-s + 655.·63-s − 595.·67-s − 211.·69-s + ⋯ |
| L(s) = 1 | − 0.397·3-s − 1.55·7-s − 0.841·9-s − 0.515·11-s − 1.85·13-s − 0.924·17-s − 0.329·19-s + 0.619·21-s + 0.925·23-s + 0.732·27-s + 0.0568·29-s + 1.58·31-s + 0.204·33-s − 0.366·37-s + 0.735·39-s − 0.948·41-s − 0.486·43-s − 1.36·47-s + 1.42·49-s + 0.367·51-s + 1.27·53-s + 0.130·57-s + 1.17·59-s − 0.855·61-s + 1.31·63-s − 1.08·67-s − 0.368·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3634297743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3634297743\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.06T + 27T^{2} \) |
| 7 | \( 1 + 28.8T + 343T^{2} \) |
| 11 | \( 1 + 18.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 64.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.87T + 2.43e4T^{2} \) |
| 31 | \( 1 - 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 82.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 595.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 569.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 435.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
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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
−9.952919025596959369688568597222, −9.147791305369012925542849577651, −8.238444069948184103130677049113, −6.98350969019151484311783447897, −6.53390851426462130587010526012, −5.43388375787446792743588980939, −4.61690275829891822816273229306, −3.12253297651345046753890979426, −2.48135633132631601826020082453, −0.31595077972304114358971938772,
0.31595077972304114358971938772, 2.48135633132631601826020082453, 3.12253297651345046753890979426, 4.61690275829891822816273229306, 5.43388375787446792743588980939, 6.53390851426462130587010526012, 6.98350969019151484311783447897, 8.238444069948184103130677049113, 9.147791305369012925542849577651, 9.952919025596959369688568597222
