| L(s) = 1 | + 2-s + 4-s − 6·7-s + 8-s − 6·14-s + 16-s + 2·25-s − 6·28-s − 4·29-s + 32-s + 8·41-s − 12·43-s + 17·49-s + 2·50-s + 20·53-s − 6·56-s − 4·58-s − 16·59-s + 64-s − 8·71-s + 22·73-s + 8·82-s − 12·86-s + 17·98-s + 2·100-s + 20·106-s + 16·107-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.26·7-s + 0.353·8-s − 1.60·14-s + 1/4·16-s + 2/5·25-s − 1.13·28-s − 0.742·29-s + 0.176·32-s + 1.24·41-s − 1.82·43-s + 17/7·49-s + 0.282·50-s + 2.74·53-s − 0.801·56-s − 0.525·58-s − 2.08·59-s + 1/8·64-s − 0.949·71-s + 2.57·73-s + 0.883·82-s − 1.29·86-s + 1.71·98-s + 1/5·100-s + 1.94·106-s + 1.54·107-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
| \(L(1)\) |
\(\approx\) |
\(1.768765991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768765991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.634785421396690321233299085650, −8.033016170548982971841517444486, −7.39951749410871353220847562437, −7.11486476424775968783668661249, −6.55362099532961595941847068962, −6.31425452124810997436354918497, −5.80584807659940157336037178503, −5.38267257036589829821591587007, −4.70810817426002516539312362730, −4.10615666775996438894538229566, −3.53039413426168323522471706796, −3.22115591001299469894859390509, −2.65093922967544299455930389544, −1.91217842788625439513224721261, −0.61251502312875154700840843574,
0.61251502312875154700840843574, 1.91217842788625439513224721261, 2.65093922967544299455930389544, 3.22115591001299469894859390509, 3.53039413426168323522471706796, 4.10615666775996438894538229566, 4.70810817426002516539312362730, 5.38267257036589829821591587007, 5.80584807659940157336037178503, 6.31425452124810997436354918497, 6.55362099532961595941847068962, 7.11486476424775968783668661249, 7.39951749410871353220847562437, 8.033016170548982971841517444486, 8.634785421396690321233299085650
