| L(s) = 1 | − 4-s − 2·5-s − 3·7-s − 6·9-s − 3·16-s + 2·20-s + 2·25-s + 3·28-s + 2·31-s + 6·35-s + 6·36-s + 14·43-s + 12·45-s + 16·61-s + 18·63-s + 7·64-s − 22·71-s + 16·73-s + 6·80-s + 27·81-s + 10·83-s − 10·89-s + 10·97-s − 2·100-s − 16·109-s + 9·112-s − 14·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1.13·7-s − 2·9-s − 3/4·16-s + 0.447·20-s + 2/5·25-s + 0.566·28-s + 0.359·31-s + 1.01·35-s + 36-s + 2.13·43-s + 1.78·45-s + 2.04·61-s + 2.26·63-s + 7/8·64-s − 2.61·71-s + 1.87·73-s + 0.670·80-s + 3·81-s + 1.09·83-s − 1.05·89-s + 1.01·97-s − 1/5·100-s − 1.53·109-s + 0.850·112-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83167 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83167 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4308512580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4308512580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 + 16 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.499325913103356853763632189381, −9.118699270520662851992946731320, −8.824077408946779925454542808940, −8.251117771973639056387663954879, −7.83738141941409087807537015447, −7.21751440221144472147737289561, −6.49232665681305663130099703764, −6.21304008848868320788005099223, −5.46839075331175164719930864289, −5.03550290063193436695283001627, −4.11816484351220827022490860162, −3.72543069880963318443598810934, −2.90686653533434217200154811654, −2.49953913974672875065937269158, −0.47723491460215022378625550922,
0.47723491460215022378625550922, 2.49953913974672875065937269158, 2.90686653533434217200154811654, 3.72543069880963318443598810934, 4.11816484351220827022490860162, 5.03550290063193436695283001627, 5.46839075331175164719930864289, 6.21304008848868320788005099223, 6.49232665681305663130099703764, 7.21751440221144472147737289561, 7.83738141941409087807537015447, 8.251117771973639056387663954879, 8.824077408946779925454542808940, 9.118699270520662851992946731320, 9.499325913103356853763632189381
