| L(s) = 1 | + 3-s − 4·4-s + 2·7-s − 2·9-s − 4·12-s + 10·13-s + 12·16-s + 4·19-s + 2·21-s + 25-s − 5·27-s − 8·28-s − 8·31-s + 8·36-s + 4·37-s + 10·39-s − 20·43-s + 12·48-s + 3·49-s − 40·52-s + 4·57-s + 16·61-s − 4·63-s − 32·64-s − 8·67-s + 4·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s + 0.755·7-s − 2/3·9-s − 1.15·12-s + 2.77·13-s + 3·16-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.962·27-s − 1.51·28-s − 1.43·31-s + 4/3·36-s + 0.657·37-s + 1.60·39-s − 3.04·43-s + 1.73·48-s + 3/7·49-s − 5.54·52-s + 0.529·57-s + 2.04·61-s − 0.503·63-s − 4·64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8948642832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8948642832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−11.50769791186352205274303435148, −10.81642134200306905635056067311, −10.26175613408033680908306725927, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.679498853252053444462430241617, −8.087023964728641191414598204496, −7.980947296320422939728903060714, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −5.47907564040983295563221596641, −4.54321060660836499326534703030, −3.61689003045514298236484607491, −3.45067056060882590557565434961, −1.35738749247200780270220785705,
1.35738749247200780270220785705, 3.45067056060882590557565434961, 3.61689003045514298236484607491, 4.54321060660836499326534703030, 5.47907564040983295563221596641, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 7.980947296320422939728903060714, 8.087023964728641191414598204496, 8.679498853252053444462430241617, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.26175613408033680908306725927, 10.81642134200306905635056067311, 11.50769791186352205274303435148
