| L(s) = 1 | + 16·2-s − 158·3-s + 256·4-s + 625·5-s − 2.52e3·6-s + 1.92e3·7-s + 4.09e3·8-s + 1.84e4·9-s + 1.00e4·10-s − 4.04e4·12-s + 3.07e4·14-s − 9.87e4·15-s + 6.55e4·16-s + 2.94e5·18-s + 1.60e5·20-s − 3.03e5·21-s + 2.11e5·23-s − 6.47e5·24-s + 3.90e5·25-s − 1.87e6·27-s + 4.92e5·28-s + 2.06e4·29-s − 1.58e6·30-s + 1.04e6·32-s + 1.20e6·35-s + 4.71e6·36-s + 2.56e6·40-s + ⋯ |
| L(s) = 1 | + 2-s − 1.95·3-s + 4-s + 5-s − 1.95·6-s + 0.800·7-s + 8-s + 2.80·9-s + 10-s − 1.95·12-s + 0.800·14-s − 1.95·15-s + 16-s + 2.80·18-s + 20-s − 1.56·21-s + 0.754·23-s − 1.95·24-s + 25-s − 3.52·27-s + 0.800·28-s + 0.0291·29-s − 1.95·30-s + 32-s + 0.800·35-s + 2.80·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.244149845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244149845\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 5 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 + 158 T + p^{8} T^{2} \) |
| 7 | \( 1 - 1922 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 - 211202 T + p^{8} T^{2} \) |
| 29 | \( 1 - 20642 T + p^{8} T^{2} \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 + 5419198 T + p^{8} T^{2} \) |
| 43 | \( 1 + 2519518 T + p^{8} T^{2} \) |
| 47 | \( 1 - 9618242 T + p^{8} T^{2} \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 + 11061598 T + p^{8} T^{2} \) |
| 67 | \( 1 + 20249758 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 + 30884638 T + p^{8} T^{2} \) |
| 89 | \( 1 + 106804798 T + p^{8} T^{2} \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−16.65350679147157346115621089045, −15.21953234335163190276189936163, −13.55779042742631229975668943981, −12.36428359379217060600695756504, −11.25671715488024942488687333267, −10.28519150065569895686899938225, −6.93167709606085666757547636599, −5.70411339529797158914340849748, −4.76336404511036741713062647681, −1.46596107424358911018234213914,
1.46596107424358911018234213914, 4.76336404511036741713062647681, 5.70411339529797158914340849748, 6.93167709606085666757547636599, 10.28519150065569895686899938225, 11.25671715488024942488687333267, 12.36428359379217060600695756504, 13.55779042742631229975668943981, 15.21953234335163190276189936163, 16.65350679147157346115621089045
