| L(s) = 1 | − 2-s − 2·7-s − 8-s − 2·11-s − 6·13-s + 2·14-s − 16-s − 4·17-s + 2·22-s − 6·23-s + 6·26-s − 10·29-s − 4·31-s + 6·32-s + 4·34-s − 4·37-s + 2·41-s + 6·43-s + 6·46-s − 4·47-s + 2·49-s − 4·53-s + 2·56-s + 10·58-s + 10·59-s + 6·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.755·7-s − 0.353·8-s − 0.603·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.426·22-s − 1.25·23-s + 1.17·26-s − 1.85·29-s − 0.718·31-s + 1.06·32-s + 0.685·34-s − 0.657·37-s + 0.312·41-s + 0.914·43-s + 0.884·46-s − 0.583·47-s + 2/7·49-s − 0.549·53-s + 0.267·56-s + 1.31·58-s + 1.30·59-s + 0.768·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 97 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 209 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.988107137886650927247968358669, −9.924995507552413765606422394654, −9.309797524028500478438069796566, −9.235179227001681562818077121714, −8.603423588712285585979841140971, −8.199124829389745687943754952952, −7.67108039533617462514262826181, −7.16056784530961885576662685931, −6.97087670327349807292039501020, −6.36037745534517283010927788543, −5.61647404196878979673117920939, −5.55035658178142213457820676917, −4.72360205450387309207626994421, −4.18723479570941456692521060289, −3.71105864611064333314274226685, −2.76553979804537263980696040981, −2.48120253506961143226195840223, −1.74952470977637072478622222235, 0, 0,
1.74952470977637072478622222235, 2.48120253506961143226195840223, 2.76553979804537263980696040981, 3.71105864611064333314274226685, 4.18723479570941456692521060289, 4.72360205450387309207626994421, 5.55035658178142213457820676917, 5.61647404196878979673117920939, 6.36037745534517283010927788543, 6.97087670327349807292039501020, 7.16056784530961885576662685931, 7.67108039533617462514262826181, 8.199124829389745687943754952952, 8.603423588712285585979841140971, 9.235179227001681562818077121714, 9.309797524028500478438069796566, 9.924995507552413765606422394654, 9.988107137886650927247968358669
