| L(s) = 1 | − 3-s − 5·9-s − 4·11-s − 2·17-s + 3·19-s + 7·25-s + 8·27-s + 4·33-s − 7·41-s + 14·43-s + 2·49-s + 2·51-s − 3·57-s + 15·59-s − 11·67-s + 5·73-s − 7·75-s + 16·81-s + 20·83-s − 2·89-s − 3·97-s + 20·99-s + 8·107-s − 18·113-s − 121-s + 7·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 5/3·9-s − 1.20·11-s − 0.485·17-s + 0.688·19-s + 7/5·25-s + 1.53·27-s + 0.696·33-s − 1.09·41-s + 2.13·43-s + 2/7·49-s + 0.280·51-s − 0.397·57-s + 1.95·59-s − 1.34·67-s + 0.585·73-s − 0.808·75-s + 16/9·81-s + 2.19·83-s − 0.211·89-s − 0.304·97-s + 2.01·99-s + 0.773·107-s − 1.69·113-s − 0.0909·121-s + 0.631·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8055363314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8055363314\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 81 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.558709272250738992173006971811, −9.046383277373142321853139360712, −8.694719447393738253918907135213, −8.127201261063222541670774482551, −7.75648920741435578347708780463, −6.99921860100790183663404446564, −6.59640418660595263690952231083, −5.83814127085546798479647792200, −5.50850356832916144606549354788, −5.07568894966195682621075966028, −4.46944353210962084762932503895, −3.44915282788138295445702095334, −2.84442635821892529867766931686, −2.31045579022302946179511932620, −0.67150016823747281771560140700,
0.67150016823747281771560140700, 2.31045579022302946179511932620, 2.84442635821892529867766931686, 3.44915282788138295445702095334, 4.46944353210962084762932503895, 5.07568894966195682621075966028, 5.50850356832916144606549354788, 5.83814127085546798479647792200, 6.59640418660595263690952231083, 6.99921860100790183663404446564, 7.75648920741435578347708780463, 8.127201261063222541670774482551, 8.694719447393738253918907135213, 9.046383277373142321853139360712, 9.558709272250738992173006971811
