| L(s) = 1 | − 1.57·2-s − 5.53·4-s + 5·5-s + 34.2·7-s + 21.2·8-s − 7.85·10-s + 26.8·11-s − 19.4·13-s − 53.7·14-s + 10.9·16-s − 29.1·17-s + 47.1·19-s − 27.6·20-s − 42.2·22-s − 113.·23-s + 25·25-s + 30.6·26-s − 189.·28-s − 81.2·29-s − 10.8·31-s − 187.·32-s + 45.7·34-s + 171.·35-s + 410.·37-s − 73.9·38-s + 106.·40-s + 443.·41-s + ⋯ |
| L(s) = 1 | − 0.555·2-s − 0.691·4-s + 0.447·5-s + 1.84·7-s + 0.939·8-s − 0.248·10-s + 0.737·11-s − 0.415·13-s − 1.02·14-s + 0.170·16-s − 0.415·17-s + 0.568·19-s − 0.309·20-s − 0.409·22-s − 1.02·23-s + 0.200·25-s + 0.230·26-s − 1.27·28-s − 0.520·29-s − 0.0627·31-s − 1.03·32-s + 0.230·34-s + 0.826·35-s + 1.82·37-s − 0.315·38-s + 0.420·40-s + 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.671094427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671094427\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 1.57T + 8T^{2} \) |
| 7 | \( 1 - 34.2T + 343T^{2} \) |
| 11 | \( 1 - 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 10.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 443.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 15.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 61.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 216.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 522.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 801.T + 9.12e5T^{2} \) |
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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
−10.85506171501565588243297746033, −9.693454184879646335557131272285, −9.097591520922642876968888215327, −8.054737368402929398075943830838, −7.54071858984230428436652956191, −5.94937748718141845084408563853, −4.86030637029309612321018450784, −4.14844552906101273265005299630, −2.07140470923719495859051141070, −0.999433502649739041219744056893,
0.999433502649739041219744056893, 2.07140470923719495859051141070, 4.14844552906101273265005299630, 4.86030637029309612321018450784, 5.94937748718141845084408563853, 7.54071858984230428436652956191, 8.054737368402929398075943830838, 9.097591520922642876968888215327, 9.693454184879646335557131272285, 10.85506171501565588243297746033
