| L(s) = 1 | − 5·5-s − 16.1·7-s + 55.5·11-s + 43.4·13-s + 25.5·17-s + 103.·19-s + 4.47·23-s + 25·25-s − 4.47·29-s + 45.1·31-s + 80.5·35-s + 69.5·37-s − 483.·41-s − 151.·43-s − 67.6·47-s − 83.6·49-s − 278.·53-s − 277.·55-s + 257.·59-s − 489.·61-s − 217.·65-s + 772.·67-s + 536.·71-s − 65.5·73-s − 894.·77-s − 749.·79-s + 1.26e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.869·7-s + 1.52·11-s + 0.926·13-s + 0.364·17-s + 1.24·19-s + 0.0405·23-s + 0.200·25-s − 0.0286·29-s + 0.261·31-s + 0.388·35-s + 0.309·37-s − 1.84·41-s − 0.538·43-s − 0.209·47-s − 0.243·49-s − 0.722·53-s − 0.680·55-s + 0.569·59-s − 1.02·61-s − 0.414·65-s + 1.40·67-s + 0.897·71-s − 0.105·73-s − 1.32·77-s − 1.06·79-s + 1.66·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.131538381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131538381\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 16.1T + 343T^{2} \) |
| 11 | \( 1 - 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.47T + 1.21e4T^{2} \) |
| 29 | \( 1 + 4.47T + 2.43e4T^{2} \) |
| 31 | \( 1 - 45.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 69.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 483.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 257.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 489.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 772.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 65.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 749.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 32.9T + 9.12e5T^{2} \) |
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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
−8.743050330787032800364311651554, −8.024798881901938031021181887621, −6.97868333190490676362477793761, −6.51319642660751826677872206699, −5.67664306302585915424190799861, −4.60019492804469337295226312184, −3.53557814764942724806714926623, −3.27512983482022174398700754904, −1.62317865142806589866639950762, −0.69993005163929511220152804932,
0.69993005163929511220152804932, 1.62317865142806589866639950762, 3.27512983482022174398700754904, 3.53557814764942724806714926623, 4.60019492804469337295226312184, 5.67664306302585915424190799861, 6.51319642660751826677872206699, 6.97868333190490676362477793761, 8.024798881901938031021181887621, 8.743050330787032800364311651554
