| L(s) = 1 | + 2.86·2-s + 0.188·4-s + 5.35·5-s − 31.0·7-s − 22.3·8-s + 15.3·10-s − 33.3·11-s − 60.8·13-s − 88.9·14-s − 65.4·16-s − 102.·17-s + 143.·19-s + 1.00·20-s − 95.4·22-s + 68.1·23-s − 96.3·25-s − 174.·26-s − 5.85·28-s − 128.·29-s + 232.·31-s − 8.52·32-s − 292.·34-s − 166.·35-s − 352.·37-s + 411.·38-s − 119.·40-s − 133.·41-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.0235·4-s + 0.478·5-s − 1.67·7-s − 0.987·8-s + 0.484·10-s − 0.914·11-s − 1.29·13-s − 1.69·14-s − 1.02·16-s − 1.45·17-s + 1.73·19-s + 0.0112·20-s − 0.924·22-s + 0.617·23-s − 0.770·25-s − 1.31·26-s − 0.0395·28-s − 0.825·29-s + 1.34·31-s − 0.0470·32-s − 1.47·34-s − 0.803·35-s − 1.56·37-s + 1.75·38-s − 0.473·40-s − 0.507·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.067632529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067632529\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.86T + 8T^{2} \) |
| 5 | \( 1 - 5.35T + 125T^{2} \) |
| 7 | \( 1 + 31.0T + 343T^{2} \) |
| 11 | \( 1 + 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 352.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 560.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 378.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 21.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 556.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−9.042561752884474602609122301310, −7.75550595766523193225911834711, −6.87752421545013213686837989993, −6.27162559511869405732177394169, −5.36723517526317218778850703377, −4.89480114056079038944412301180, −3.75269116303281015989003505735, −2.96055418227611519837462119864, −2.35317560173020367997020720024, −0.37015152582664798335490258948,
0.37015152582664798335490258948, 2.35317560173020367997020720024, 2.96055418227611519837462119864, 3.75269116303281015989003505735, 4.89480114056079038944412301180, 5.36723517526317218778850703377, 6.27162559511869405732177394169, 6.87752421545013213686837989993, 7.75550595766523193225911834711, 9.042561752884474602609122301310
