| L(s) = 1 | − 3-s − 4-s − 7-s − 2·9-s + 12-s − 7·13-s + 16-s − 10·19-s + 21-s + 6·25-s + 5·27-s + 28-s + 9·31-s + 2·36-s − 19·37-s + 7·39-s − 2·43-s − 48-s − 4·49-s + 7·52-s + 10·57-s + 8·61-s + 2·63-s − 64-s − 4·67-s + 3·73-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 − 2.29·19-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 − 3.12·37-s + 1.12·39-s − 0.304·43-s − 0.144·48-s − 4/7·49-s + 0.970·52-s + 1.32·57-s + 1.02·61-s + 0.251·63-s − 1/8·64-s − 0.488·67-s + 0.351·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15372 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15372 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 7 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.69933606551496377262993476555, −10.27646993101050404713291917330, −9.986404121043595563545103186090, −9.082824861062935926477384595637, −8.644814175485903451584336313747, −8.246954723425298148564770219772, −7.33315693541240693525865236486, −6.61721451700289991026263077112, −6.39095729875628068094057281928, −5.25838169655916773107906224511, −4.98300637659469421429940775764, −4.25821083407719748370965090820, −3.14938292389174939671609465588, −2.27543300626313850221028928504, 0,
2.27543300626313850221028928504, 3.14938292389174939671609465588, 4.25821083407719748370965090820, 4.98300637659469421429940775764, 5.25838169655916773107906224511, 6.39095729875628068094057281928, 6.61721451700289991026263077112, 7.33315693541240693525865236486, 8.246954723425298148564770219772, 8.644814175485903451584336313747, 9.082824861062935926477384595637, 9.986404121043595563545103186090, 10.27646993101050404713291917330, 10.69933606551496377262993476555
