| L(s) = 1 | + 3.18·2-s − 5.56·3-s + 2.14·4-s − 0.334·5-s − 17.7·6-s − 18.6·8-s + 3.95·9-s − 1.06·10-s − 38.4·11-s − 11.9·12-s − 84.0·13-s + 1.86·15-s − 76.5·16-s − 95.0·17-s + 12.5·18-s − 31.7·19-s − 0.719·20-s − 122.·22-s + 119.·23-s + 103.·24-s − 124.·25-s − 267.·26-s + 128.·27-s − 134.·29-s + 5.93·30-s − 320.·31-s − 94.8·32-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 1.07·3-s + 0.268·4-s − 0.0299·5-s − 1.20·6-s − 0.823·8-s + 0.146·9-s − 0.0337·10-s − 1.05·11-s − 0.287·12-s − 1.79·13-s + 0.0320·15-s − 1.19·16-s − 1.35·17-s + 0.164·18-s − 0.383·19-s − 0.00804·20-s − 1.18·22-s + 1.08·23-s + 0.882·24-s − 0.999·25-s − 2.01·26-s + 0.913·27-s − 0.864·29-s + 0.0361·30-s − 1.85·31-s − 0.523·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.009268277813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009268277813\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 - 3.18T + 8T^{2} \) |
| 3 | \( 1 + 5.56T + 27T^{2} \) |
| 5 | \( 1 + 0.334T + 125T^{2} \) |
| 11 | \( 1 + 38.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 320.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 347.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 253.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 30.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 156.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 710.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 414.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 79.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 227.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 17.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 350.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 201.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 366.T + 9.12e5T^{2} \) |
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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
−8.946220860614451375665845313155, −7.72494234461705316979722474520, −6.99425328114545904444429387823, −6.10579767767270040636468235971, −5.40448170417250716681940451848, −4.89434871095350394777928918629, −4.23670947119062597288920645157, −2.96090836070646469411223927969, −2.17666037804478335492073857507, −0.03607702679783432181928154151,
0.03607702679783432181928154151, 2.17666037804478335492073857507, 2.96090836070646469411223927969, 4.23670947119062597288920645157, 4.89434871095350394777928918629, 5.40448170417250716681940451848, 6.10579767767270040636468235971, 6.99425328114545904444429387823, 7.72494234461705316979722474520, 8.946220860614451375665845313155
