| L(s) = 1 | + 3.48e9·3-s − 1.53e17·7-s + 1.21e19·9-s + 9.24e21·13-s − 6.04e25·19-s − 5.35e26·21-s + 9.09e27·25-s + 4.23e28·27-s − 7.13e29·31-s + 2.88e31·37-s + 3.22e31·39-s + 4.15e32·43-s + 1.72e34·49-s − 2.10e35·57-s + 9.08e35·61-s − 1.86e36·63-s − 4.15e36·67-s + 3.66e37·73-s + 3.17e37·75-s − 1.68e38·79-s + 1.47e38·81-s − 1.42e39·91-s − 2.48e39·93-s + 1.06e40·97-s + 3.02e40·103-s − 8.60e40·109-s + 1.00e41·111-s + ⋯ |
| L(s) = 1 | + 3-s − 1.92·7-s + 9-s + 0.486·13-s − 1.60·19-s − 1.92·21-s + 25-s + 27-s − 1.06·31-s + 1.24·37-s + 0.486·39-s + 0.890·43-s + 2.70·49-s − 1.60·57-s + 1.78·61-s − 1.92·63-s − 1.24·67-s + 1.98·73-s + 75-s − 1.88·79-s + 81-s − 0.936·91-s − 1.06·93-s + 1.95·97-s + 1.67·103-s − 1.53·109-s + 1.24·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(2.345420889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345420889\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{20} T \) |
| good | 5 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 7 | \( 1 + 153603710655044926 T + p^{40} T^{2} \) |
| 11 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 13 | \( 1 - \)\(92\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 17 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 19 | \( 1 + \)\(60\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 23 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 29 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 31 | \( 1 + \)\(71\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 37 | \( 1 - \)\(28\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 41 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 43 | \( 1 - \)\(41\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 47 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 53 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 59 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 61 | \( 1 - \)\(90\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 67 | \( 1 + \)\(41\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 71 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 73 | \( 1 - \)\(36\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 79 | \( 1 + \)\(16\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 83 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 89 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 97 | \( 1 - \)\(10\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.70443636406415483097370862808, −10.59392105539111837073752758189, −9.501714511145044957051130473286, −8.621342814620769194561215850818, −7.08577810755957694504058158242, −6.14034133798700448832873567319, −4.15522345456365657385739977781, −3.23051199227731252943438583099, −2.25690632854127484514899987968, −0.65594745625766533477890050437,
0.65594745625766533477890050437, 2.25690632854127484514899987968, 3.23051199227731252943438583099, 4.15522345456365657385739977781, 6.14034133798700448832873567319, 7.08577810755957694504058158242, 8.621342814620769194561215850818, 9.501714511145044957051130473286, 10.59392105539111837073752758189, 12.70443636406415483097370862808
