| L(s) = 1 | + 5.56·2-s + 3.56·3-s + 22.9·4-s + 19.8·6-s − 6.05·7-s + 83.0·8-s − 14.3·9-s − 11·11-s + 81.6·12-s + 4.38·13-s − 33.6·14-s + 278.·16-s + 110.·17-s − 79.6·18-s − 94.2·19-s − 21.5·21-s − 61.1·22-s − 15.7·23-s + 295.·24-s + 24.3·26-s − 147.·27-s − 138.·28-s − 256.·29-s − 170.·31-s + 883.·32-s − 39.1·33-s + 614.·34-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.685·3-s + 2.86·4-s + 1.34·6-s − 0.326·7-s + 3.66·8-s − 0.530·9-s − 0.301·11-s + 1.96·12-s + 0.0935·13-s − 0.642·14-s + 4.34·16-s + 1.57·17-s − 1.04·18-s − 1.13·19-s − 0.224·21-s − 0.592·22-s − 0.142·23-s + 2.51·24-s + 0.183·26-s − 1.04·27-s − 0.936·28-s − 1.64·29-s − 0.988·31-s + 4.88·32-s − 0.206·33-s + 3.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.147402378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.147402378\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.56T + 8T^{2} \) |
| 3 | \( 1 - 3.56T + 27T^{2} \) |
| 7 | \( 1 + 6.05T + 343T^{2} \) |
| 13 | \( 1 - 4.38T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 282.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 167.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 919.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 882.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 277.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 977.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−11.69572016605328762640258971878, −10.94642779717614699350595196286, −9.791117252549633231185960275416, −8.178086329002355171538578448866, −7.29460882652785275914456489596, −6.05652779798234456546893902156, −5.34495203944260822115612825842, −3.93093175354559237603576379166, −3.15864310528863240940297641906, −2.03656860698502110908573076507,
2.03656860698502110908573076507, 3.15864310528863240940297641906, 3.93093175354559237603576379166, 5.34495203944260822115612825842, 6.05652779798234456546893902156, 7.29460882652785275914456489596, 8.178086329002355171538578448866, 9.791117252549633231185960275416, 10.94642779717614699350595196286, 11.69572016605328762640258971878
