| L(s) = 1 | − 3-s + 4-s − 7-s − 2·9-s − 12-s − 7·13-s + 16-s − 5·19-s + 21-s + 6·25-s + 5·27-s − 28-s + 9·31-s − 2·36-s + 11·37-s + 7·39-s + 8·43-s − 48-s − 4·49-s − 7·52-s + 5·57-s + 9·61-s + 2·63-s + 64-s − 4·67-s − 17·73-s − 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.377·7-s − 2/3·9-s − 0.288·12-s − 1.94·13-s + 1/4·16-s − 1.14·19-s + 0.218·21-s + 6/5·25-s + 0.962·27-s − 0.188·28-s + 1.61·31-s − 1/3·36-s + 1.80·37-s + 1.12·39-s + 1.21·43-s − 0.144·48-s − 4/7·49-s − 0.970·52-s + 0.662·57-s + 1.15·61-s + 0.251·63-s + 1/8·64-s − 0.488·67-s − 1.98·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 1789 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 35 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.226085974067293901489389387753, −7.970483615512785459025261827594, −7.28200305284809743740153090903, −6.93095005547487451682702625467, −6.53061929978045539841318740488, −6.01274754493439557880388918999, −5.66486294397242041190297807646, −5.01005105923968944108816895455, −4.51905569924110223277962555624, −4.20600752876485526348008109879, −3.08831035841540254630056600763, −2.64544491067746526307655911863, −2.38513133002422672176234268536, −1.09792060193784864885850589753, 0,
1.09792060193784864885850589753, 2.38513133002422672176234268536, 2.64544491067746526307655911863, 3.08831035841540254630056600763, 4.20600752876485526348008109879, 4.51905569924110223277962555624, 5.01005105923968944108816895455, 5.66486294397242041190297807646, 6.01274754493439557880388918999, 6.53061929978045539841318740488, 6.93095005547487451682702625467, 7.28200305284809743740153090903, 7.970483615512785459025261827594, 8.226085974067293901489389387753
