| L(s) = 1 | − 2·4-s − 8·11-s − 5·16-s − 8·19-s − 8·29-s + 24·31-s + 8·41-s + 16·44-s − 2·49-s − 8·61-s + 20·64-s − 40·71-s + 16·76-s − 16·79-s − 8·89-s + 56·101-s + 8·109-s + 16·116-s + 36·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s − 2.41·11-s − 5/4·16-s − 1.83·19-s − 1.48·29-s + 4.31·31-s + 1.24·41-s + 2.41·44-s − 2/7·49-s − 1.02·61-s + 5/2·64-s − 4.74·71-s + 1.83·76-s − 1.80·79-s − 0.847·89-s + 5.57·101-s + 0.766·109-s + 1.48·116-s + 3.27·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1754483824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1754483824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 698 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9382 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 6218 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 42502 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.74838916203258435424600027745, −6.30026191335359124632248968138, −6.20308931677202613031965305342, −6.13826055286665482763449939279, −5.87063825595099053340774904441, −5.66767158298693579228743521215, −5.53079713591420088921059849880, −4.89732556690609095038396281743, −4.83226894706813712562199591532, −4.69514735753630666949375857691, −4.65841611519778438721181008705, −4.42666281802822174247804346152, −4.17970849162196801935524924637, −3.93818049270252831430957372544, −3.49030236416977741467500677624, −3.18642491838913563963657999735, −3.09391111446318855459571571870, −2.54547368211328641730225610703, −2.46766060647145738656799321563, −2.34102060443871521469960945171, −2.14114308615981878430954931506, −1.33320315469056512678066031339, −1.32632172643212038205313858749, −0.52542446725463026152725304473, −0.12944883863079460234655042444,
0.12944883863079460234655042444, 0.52542446725463026152725304473, 1.32632172643212038205313858749, 1.33320315469056512678066031339, 2.14114308615981878430954931506, 2.34102060443871521469960945171, 2.46766060647145738656799321563, 2.54547368211328641730225610703, 3.09391111446318855459571571870, 3.18642491838913563963657999735, 3.49030236416977741467500677624, 3.93818049270252831430957372544, 4.17970849162196801935524924637, 4.42666281802822174247804346152, 4.65841611519778438721181008705, 4.69514735753630666949375857691, 4.83226894706813712562199591532, 4.89732556690609095038396281743, 5.53079713591420088921059849880, 5.66767158298693579228743521215, 5.87063825595099053340774904441, 6.13826055286665482763449939279, 6.20308931677202613031965305342, 6.30026191335359124632248968138, 6.74838916203258435424600027745
