| L(s) = 1 | + 6·3-s − 3·4-s + 16·7-s + 27·9-s − 18·12-s − 8·13-s − 7·16-s + 12·19-s + 96·21-s + 6·25-s + 108·27-s − 48·28-s − 52·31-s − 81·36-s + 60·37-s − 48·39-s − 84·43-s − 42·48-s + 94·49-s + 24·52-s + 72·57-s − 24·61-s + 432·63-s + 69·64-s + 4·67-s + 148·73-s + 36·75-s + ⋯ |
| L(s) = 1 | + 2·3-s − 3/4·4-s + 16/7·7-s + 3·9-s − 3/2·12-s − 0.615·13-s − 0.437·16-s + 0.631·19-s + 32/7·21-s + 6/25·25-s + 4·27-s − 1.71·28-s − 1.67·31-s − 9/4·36-s + 1.62·37-s − 1.23·39-s − 1.95·43-s − 7/8·48-s + 1.91·49-s + 6/13·52-s + 1.26·57-s − 0.393·61-s + 48/7·63-s + 1.07·64-s + 4/67·67-s + 2.02·73-s + 0.479·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.678511298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.678511298\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1014 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 98 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3186 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3018 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6518 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12194 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1586 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.57004167915732868983528299016, −10.85938176637635855243928737379, −10.51396447632304208356989650948, −9.726769499814129162082651552549, −9.525196202594489687419529564589, −9.102348159074849988871042884137, −8.421573525065110104475712916289, −8.390712779255600836541277311472, −7.83190707614531253158941923857, −7.42646468162204052913179520957, −7.10510599803147858989487005421, −6.22539578064342916048976025464, −5.05633495519541799269384137125, −4.92557891552064406219353494769, −4.52576966695752003617016637773, −3.78399226838600941140469186207, −3.28550989201391704616514474527, −2.22922139084358807648094135907, −1.95828073273500628633196463102, −1.08156298363179726751424690256,
1.08156298363179726751424690256, 1.95828073273500628633196463102, 2.22922139084358807648094135907, 3.28550989201391704616514474527, 3.78399226838600941140469186207, 4.52576966695752003617016637773, 4.92557891552064406219353494769, 5.05633495519541799269384137125, 6.22539578064342916048976025464, 7.10510599803147858989487005421, 7.42646468162204052913179520957, 7.83190707614531253158941923857, 8.390712779255600836541277311472, 8.421573525065110104475712916289, 9.102348159074849988871042884137, 9.525196202594489687419529564589, 9.726769499814129162082651552549, 10.51396447632304208356989650948, 10.85938176637635855243928737379, 11.57004167915732868983528299016
