| L(s) = 1 | − 48·5-s + 81·9-s − 240·13-s − 322·17-s + 1.67e3·25-s + 1.68e3·29-s + 1.68e3·37-s + 3.03e3·41-s − 3.88e3·45-s + 2.40e3·49-s − 5.04e3·53-s + 2.64e3·61-s + 1.15e4·65-s + 1.44e3·73-s + 6.56e3·81-s + 1.54e4·85-s − 9.75e3·89-s + 1.91e3·97-s − 7.92e3·101-s + 2.18e4·109-s − 2.46e4·113-s − 1.94e4·117-s + ⋯ |
| L(s) = 1 | − 1.91·5-s + 9-s − 1.42·13-s − 1.11·17-s + 2.68·25-s + 1.99·29-s + 1.22·37-s + 1.80·41-s − 1.91·45-s + 49-s − 1.79·53-s + 0.709·61-s + 2.72·65-s + 0.270·73-s + 81-s + 2.13·85-s − 1.23·89-s + 0.203·97-s − 0.776·101-s + 1.83·109-s − 1.92·113-s − 1.42·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.074139298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074139298\) |
| \(L(3)\) |
|
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 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( 1 + 48 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 240 T + p^{4} T^{2} \) |
| 17 | \( 1 + 322 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 - 1680 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 - 1680 T + p^{4} T^{2} \) |
| 41 | \( 1 - 3038 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 5040 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 2640 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 1442 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 9758 T + p^{4} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{4} T^{2} \) |
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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
−11.44580608115986296302274755540, −10.56863214378507404409860742225, −9.417651606876977011600211138497, −8.215393936559517856644068789814, −7.45946971373675188662286729007, −6.70463588586799736586589052926, −4.67136260700257677066105519321, −4.22566529978624045135258582538, −2.72536697854778735710659457663, −0.65356341246446037616491421322,
0.65356341246446037616491421322, 2.72536697854778735710659457663, 4.22566529978624045135258582538, 4.67136260700257677066105519321, 6.70463588586799736586589052926, 7.45946971373675188662286729007, 8.215393936559517856644068789814, 9.417651606876977011600211138497, 10.56863214378507404409860742225, 11.44580608115986296302274755540
