| L(s) = 1 | + 8·3-s − 10·5-s + 16·7-s + 37·9-s − 40·11-s − 50·13-s − 80·15-s − 30·17-s + 40·19-s + 128·21-s + 48·23-s − 25·25-s + 80·27-s − 34·29-s + 320·31-s − 320·33-s − 160·35-s + 310·37-s − 400·39-s + 410·41-s + 152·43-s − 370·45-s − 416·47-s − 87·49-s − 240·51-s − 410·53-s + 400·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.894·5-s + 0.863·7-s + 1.37·9-s − 1.09·11-s − 1.06·13-s − 1.37·15-s − 0.428·17-s + 0.482·19-s + 1.33·21-s + 0.435·23-s − 1/5·25-s + 0.570·27-s − 0.217·29-s + 1.85·31-s − 1.68·33-s − 0.772·35-s + 1.37·37-s − 1.64·39-s + 1.56·41-s + 0.539·43-s − 1.22·45-s − 1.29·47-s − 0.253·49-s − 0.658·51-s − 1.06·53-s + 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.647789167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.647789167\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 - 320 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 416 T + p^{3} T^{2} \) |
| 53 | \( 1 + 410 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 776 T + p^{3} T^{2} \) |
| 71 | \( 1 - 400 T + p^{3} T^{2} \) |
| 73 | \( 1 + 630 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 - 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−15.85249179829585567904336618178, −15.03563193490656579912823293464, −14.13504791546597104644057823567, −12.86927620499612524351220202170, −11.35227830488608030471932546259, −9.682824502714614814012798336544, −8.163652281493415771325020581764, −7.61350884803859455478884620237, −4.56474133618286482949667896527, −2.69410710716516310854206677123,
2.69410710716516310854206677123, 4.56474133618286482949667896527, 7.61350884803859455478884620237, 8.163652281493415771325020581764, 9.682824502714614814012798336544, 11.35227830488608030471932546259, 12.86927620499612524351220202170, 14.13504791546597104644057823567, 15.03563193490656579912823293464, 15.85249179829585567904336618178
