| L(s) = 1 | + 2·2-s + 4·4-s + 22·5-s + 8·8-s + 44·10-s − 26·11-s + 54·13-s + 16·16-s + 74·17-s − 116·19-s + 88·20-s − 52·22-s + 58·23-s + 359·25-s + 108·26-s − 208·29-s + 252·31-s + 32·32-s + 148·34-s + 50·37-s − 232·38-s + 176·40-s + 126·41-s + 164·43-s − 104·44-s + 116·46-s − 444·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.96·5-s + 0.353·8-s + 1.39·10-s − 0.712·11-s + 1.15·13-s + 1/4·16-s + 1.05·17-s − 1.40·19-s + 0.983·20-s − 0.503·22-s + 0.525·23-s + 2.87·25-s + 0.814·26-s − 1.33·29-s + 1.46·31-s + 0.176·32-s + 0.746·34-s + 0.222·37-s − 0.990·38-s + 0.695·40-s + 0.479·41-s + 0.581·43-s − 0.356·44-s + 0.371·46-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.117002753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.117002753\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 208 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 50 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 444 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 T + p^{3} T^{2} \) |
| 59 | \( 1 - 124 T + p^{3} T^{2} \) |
| 61 | \( 1 - 162 T + p^{3} T^{2} \) |
| 67 | \( 1 + 860 T + p^{3} T^{2} \) |
| 71 | \( 1 - 238 T + p^{3} T^{2} \) |
| 73 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 984 T + p^{3} T^{2} \) |
| 83 | \( 1 - 656 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 + 526 T + p^{3} T^{2} \) |
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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
−9.928447354829510665878244833224, −9.014101934452921842812626781747, −8.090706976205785545259873055285, −6.80427958556290181180303829288, −6.01554130010842263619846464606, −5.57816309483299601932902155705, −4.55954179008165913926209813787, −3.17126603812081939439546712567, −2.22940218346086583949013049125, −1.24928973323384161399241151456,
1.24928973323384161399241151456, 2.22940218346086583949013049125, 3.17126603812081939439546712567, 4.55954179008165913926209813787, 5.57816309483299601932902155705, 6.01554130010842263619846464606, 6.80427958556290181180303829288, 8.090706976205785545259873055285, 9.014101934452921842812626781747, 9.928447354829510665878244833224
