| L(s) = 1 | + 1.04e6·2-s − 3.07e9·3-s + 1.09e12·4-s + 9.53e13·5-s − 3.22e15·6-s + 1.40e17·7-s + 1.15e18·8-s − 2.67e18·9-s + 1.00e20·10-s − 3.38e21·12-s + 1.47e23·14-s − 2.93e23·15-s + 1.20e24·16-s − 2.80e24·18-s + 1.04e26·20-s − 4.34e26·21-s + 3.20e27·23-s − 3.55e27·24-s + 9.09e27·25-s + 4.56e28·27-s + 1.54e29·28-s + 2.58e28·29-s − 3.07e29·30-s + 1.26e30·32-s + 1.34e31·35-s − 2.94e30·36-s + 1.09e32·40-s + ⋯ |
| L(s) = 1 | + 2-s − 0.883·3-s + 4-s + 5-s − 0.883·6-s + 1.76·7-s + 8-s − 0.219·9-s + 10-s − 0.883·12-s + 1.76·14-s − 0.883·15-s + 16-s − 0.219·18-s + 20-s − 1.56·21-s + 1.86·23-s − 0.883·24-s + 25-s + 1.07·27-s + 1.76·28-s + 0.145·29-s − 0.883·30-s + 32-s + 1.76·35-s − 0.219·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(6.124989066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.124989066\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{20} T \) |
| 5 | \( 1 - p^{20} T \) |
| good | 3 | \( 1 + 3079559198 T + p^{40} T^{2} \) |
| 7 | \( 1 - 140945283647952002 T + p^{40} T^{2} \) |
| 11 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 13 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 17 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 19 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 23 | \( 1 - \)\(32\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 29 | \( 1 - \)\(25\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 31 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 37 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 41 | \( 1 + \)\(47\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 43 | \( 1 + \)\(88\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 47 | \( 1 - \)\(36\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 53 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 59 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 61 | \( 1 + \)\(90\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 67 | \( 1 + \)\(32\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 71 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 73 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 79 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 83 | \( 1 + \)\(47\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 89 | \( 1 - \)\(18\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 97 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
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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
−11.18536404707102087249392800609, −10.51633553961312359808605426157, −8.660630018752931862545576616922, −7.22061812053256996678948263134, −6.06431764060825460682629797729, −5.15604294075701661295346733300, −4.71067717368089089569519505926, −2.92926050943879108892260731925, −1.78624962738186755109003512476, −1.00981357809695663977486863217,
1.00981357809695663977486863217, 1.78624962738186755109003512476, 2.92926050943879108892260731925, 4.71067717368089089569519505926, 5.15604294075701661295346733300, 6.06431764060825460682629797729, 7.22061812053256996678948263134, 8.660630018752931862545576616922, 10.51633553961312359808605426157, 11.18536404707102087249392800609
