| L(s) = 1 | − 2·4-s − 2·9-s − 20·11-s + 3·16-s + 20·19-s − 8·25-s + 8·31-s + 4·36-s − 32·41-s + 40·44-s + 22·49-s + 24·59-s + 8·61-s − 4·64-s − 32·71-s − 40·76-s − 32·79-s + 3·81-s + 28·89-s + 40·99-s + 16·100-s + 24·101-s − 36·109-s + 210·121-s − 16·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s − 6.03·11-s + 3/4·16-s + 4.58·19-s − 8/5·25-s + 1.43·31-s + 2/3·36-s − 4.99·41-s + 6.03·44-s + 22/7·49-s + 3.12·59-s + 1.02·61-s − 1/2·64-s − 3.79·71-s − 4.58·76-s − 3.60·79-s + 1/3·81-s + 2.96·89-s + 4.02·99-s + 8/5·100-s + 2.38·101-s − 3.44·109-s + 19.0·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2507670081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2507670081\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 315 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 427 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 62 p T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 194 T^{2} + 14995 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 23734 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 226 T^{2} + 22627 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 118 T^{2} + 13731 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 4494 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.20304895068573087182937466285, −6.99255461976536827688969365086, −6.69640481852430767602328493362, −6.32793177543000704781684784022, −5.75028243834403966390769203499, −5.70072068444289118259050456784, −5.56494869353098323184789162774, −5.38773032673114205143703896661, −5.23976147055750973998115521099, −5.23158957079379639123550981183, −4.95383299185688125703484283755, −4.64455808985397001753157061976, −4.49686991598703664509884565010, −3.84722069413717081328236694406, −3.72706007252514306689074627160, −3.33398058283875100785931057948, −3.17461469679749977336425181653, −2.82872287498152258370242058337, −2.69575198636421418207435831836, −2.61178069832126933487519023709, −2.19457354981061130362456614644, −1.70346025938810581967365430448, −1.17396202243104779597542778735, −0.62232481732299538339438259789, −0.17281953492235326151670769701,
0.17281953492235326151670769701, 0.62232481732299538339438259789, 1.17396202243104779597542778735, 1.70346025938810581967365430448, 2.19457354981061130362456614644, 2.61178069832126933487519023709, 2.69575198636421418207435831836, 2.82872287498152258370242058337, 3.17461469679749977336425181653, 3.33398058283875100785931057948, 3.72706007252514306689074627160, 3.84722069413717081328236694406, 4.49686991598703664509884565010, 4.64455808985397001753157061976, 4.95383299185688125703484283755, 5.23158957079379639123550981183, 5.23976147055750973998115521099, 5.38773032673114205143703896661, 5.56494869353098323184789162774, 5.70072068444289118259050456784, 5.75028243834403966390769203499, 6.32793177543000704781684784022, 6.69640481852430767602328493362, 6.99255461976536827688969365086, 7.20304895068573087182937466285
