| L(s) = 1 | + 2-s − 4-s − 3·8-s + 2·11-s − 8·13-s − 16-s + 2·22-s + 2·23-s + 6·25-s − 8·26-s + 5·32-s + 4·37-s − 2·44-s + 2·46-s − 6·47-s − 2·49-s + 6·50-s + 8·52-s + 8·59-s + 8·61-s + 7·64-s − 12·71-s − 4·73-s + 4·74-s − 18·83-s − 6·88-s − 2·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.603·11-s − 2.21·13-s − 1/4·16-s + 0.426·22-s + 0.417·23-s + 6/5·25-s − 1.56·26-s + 0.883·32-s + 0.657·37-s − 0.301·44-s + 0.294·46-s − 0.875·47-s − 2/7·49-s + 0.848·50-s + 1.10·52-s + 1.04·59-s + 1.02·61-s + 7/8·64-s − 1.42·71-s − 0.468·73-s + 0.464·74-s − 1.97·83-s − 0.639·88-s − 0.208·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.369655268501135718878559418172, −7.86711974735970994860599630971, −7.29830961204115281346571304271, −6.88021464572161832766849052602, −6.57085385001364899245096014520, −5.81726961340408254214003936689, −5.41037292108441356286935825951, −4.89327426123340817894900461903, −4.58037201218432950194739519485, −4.10263146164599737267263727065, −3.38461953868257182046646445127, −2.79713793837996014723286792599, −2.37095502353446026828469456471, −1.20500386288787110349871315562, 0,
1.20500386288787110349871315562, 2.37095502353446026828469456471, 2.79713793837996014723286792599, 3.38461953868257182046646445127, 4.10263146164599737267263727065, 4.58037201218432950194739519485, 4.89327426123340817894900461903, 5.41037292108441356286935825951, 5.81726961340408254214003936689, 6.57085385001364899245096014520, 6.88021464572161832766849052602, 7.29830961204115281346571304271, 7.86711974735970994860599630971, 8.369655268501135718878559418172
