| L(s) = 1 | + 8·3-s + 14·5-s + 7·7-s + 37·9-s − 28·11-s − 18·13-s + 112·15-s + 74·17-s + 80·19-s + 56·21-s + 112·23-s + 71·25-s + 80·27-s − 190·29-s − 72·31-s − 224·33-s + 98·35-s + 346·37-s − 144·39-s + 162·41-s − 412·43-s + 518·45-s − 24·47-s + 49·49-s + 592·51-s − 318·53-s − 392·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 1.25·5-s + 0.377·7-s + 1.37·9-s − 0.767·11-s − 0.384·13-s + 1.92·15-s + 1.05·17-s + 0.965·19-s + 0.581·21-s + 1.01·23-s + 0.567·25-s + 0.570·27-s − 1.21·29-s − 0.417·31-s − 1.18·33-s + 0.473·35-s + 1.53·37-s − 0.591·39-s + 0.617·41-s − 1.46·43-s + 1.71·45-s − 0.0744·47-s + 1/7·49-s + 1.62·51-s − 0.824·53-s − 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.283974960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.283974960\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 + 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−10.29633545839499744933870815156, −9.605566722010257815589731278328, −9.053261234170919809751656970241, −7.904554768202042088224548986474, −7.37375290430172677603737764982, −5.85006072957986524555581705593, −4.93783003046888888408441198634, −3.36264239617455678497994752113, −2.50788201354696195983896571886, −1.47189618084367035279294653513,
1.47189618084367035279294653513, 2.50788201354696195983896571886, 3.36264239617455678497994752113, 4.93783003046888888408441198634, 5.85006072957986524555581705593, 7.37375290430172677603737764982, 7.904554768202042088224548986474, 9.053261234170919809751656970241, 9.605566722010257815589731278328, 10.29633545839499744933870815156
