| L(s) = 1 | + 4·2-s + 16·4-s − 98·7-s + 64·8-s − 354·11-s − 404·13-s − 392·14-s + 256·16-s + 654·17-s + 1.79e3·19-s − 1.41e3·22-s − 1.08e3·23-s − 1.61e3·26-s − 1.56e3·28-s + 5.75e3·29-s + 1.01e4·31-s + 1.02e3·32-s + 2.61e3·34-s − 5.55e3·37-s + 7.18e3·38-s + 1.29e4·41-s + 8.96e3·43-s − 5.66e3·44-s − 4.32e3·46-s − 5.40e3·47-s − 7.20e3·49-s − 6.46e3·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.882·11-s − 0.663·13-s − 0.534·14-s + 1/4·16-s + 0.548·17-s + 1.14·19-s − 0.623·22-s − 0.425·23-s − 0.468·26-s − 0.377·28-s + 1.27·29-s + 1.90·31-s + 0.176·32-s + 0.388·34-s − 0.666·37-s + 0.807·38-s + 1.20·41-s + 0.739·43-s − 0.441·44-s − 0.301·46-s − 0.356·47-s − 3/7·49-s − 0.331·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.907107739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907107739\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 + 354 T + p^{5} T^{2} \) |
| 13 | \( 1 + 404 T + p^{5} T^{2} \) |
| 17 | \( 1 - 654 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1796 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1080 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5754 T + p^{5} T^{2} \) |
| 31 | \( 1 - 10196 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5552 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12960 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8968 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5400 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8214 T + p^{5} T^{2} \) |
| 59 | \( 1 + 3954 T + p^{5} T^{2} \) |
| 61 | \( 1 - 962 T + p^{5} T^{2} \) |
| 67 | \( 1 - 4 p^{2} T + p^{5} T^{2} \) |
| 71 | \( 1 - 56148 T + p^{5} T^{2} \) |
| 73 | \( 1 - 85690 T + p^{5} T^{2} \) |
| 79 | \( 1 + 26044 T + p^{5} T^{2} \) |
| 83 | \( 1 + 93468 T + p^{5} T^{2} \) |
| 89 | \( 1 + 73428 T + p^{5} T^{2} \) |
| 97 | \( 1 + 128978 T + p^{5} T^{2} \) |
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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
−10.15861891137542177656946007964, −9.737442556441800431680139330262, −8.278098413470103190138671520528, −7.41688853671107076566945530638, −6.44141510138619555207319971023, −5.47740466032390459204341053072, −4.57454826371485185044299013874, −3.25991126734082163455883651786, −2.49828633736484019382719879837, −0.78102026876971192018029879246,
0.78102026876971192018029879246, 2.49828633736484019382719879837, 3.25991126734082163455883651786, 4.57454826371485185044299013874, 5.47740466032390459204341053072, 6.44141510138619555207319971023, 7.41688853671107076566945530638, 8.278098413470103190138671520528, 9.737442556441800431680139330262, 10.15861891137542177656946007964
