| L(s) = 1 | − 3·5-s − 6·17-s + 6·19-s + 30·23-s + 9·25-s + 16·31-s − 48·47-s + 137·49-s + 192·53-s + 38·61-s − 6·79-s + 288·83-s + 18·85-s − 18·95-s − 18·107-s − 226·109-s + 564·113-s − 90·115-s + 129·121-s − 102·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + ⋯ |
| L(s) = 1 | − 3/5·5-s − 0.352·17-s + 6/19·19-s + 1.30·23-s + 9/25·25-s + 0.516·31-s − 1.02·47-s + 2.79·49-s + 3.62·53-s + 0.622·61-s − 0.0759·79-s + 3.46·83-s + 0.211·85-s − 0.189·95-s − 0.168·107-s − 2.07·109-s + 4.99·113-s − 0.782·115-s + 1.06·121-s − 0.815·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.309·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.518182615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.518182615\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 137 T^{2} + 9024 T^{4} - 137 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 129 T^{2} + 10400 T^{4} - 129 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 136 T^{2} - 5970 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 528 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 3 T + 254 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 15 T + 1062 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 57 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1522 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3186 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 776 T^{2} + 5716590 T^{4} - 776 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 24 T + 3726 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 96 T + 6041 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7044 T^{2} + 28934630 T^{4} - 7044 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 19 T + 7062 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6584 T^{2} + 19350606 T^{4} - 6584 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 3 T + 8252 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 144 T + 13737 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1879 T^{2} + 54186000 T^{4} + 1879 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−7.59480249378731988578623493736, −7.24486505499314037530679437588, −7.14965074264055476362853282340, −7.01275670838954465020833305254, −6.65695917651390382627637725198, −6.33605370284512729798904460401, −6.31335588900175068848086073253, −5.70298091042920946086717120353, −5.58384010389784811522242015628, −5.51443902568099902141323608903, −4.93343453075699766950193652077, −4.91146452031773010622554800347, −4.62564200572480354485948660772, −4.26250676419046964951487041762, −3.82051151121896728239863977875, −3.80559388466228488940765137014, −3.57492943798616416539456564058, −2.98460700410702451579507870099, −2.79429472960029197822341603246, −2.54622887259097882223733260228, −1.95837056112683960621635953541, −1.92196306624180985342596641615, −1.02290069279573665269526890177, −0.76760870114311455137911986603, −0.54311710254509418368073126319,
0.54311710254509418368073126319, 0.76760870114311455137911986603, 1.02290069279573665269526890177, 1.92196306624180985342596641615, 1.95837056112683960621635953541, 2.54622887259097882223733260228, 2.79429472960029197822341603246, 2.98460700410702451579507870099, 3.57492943798616416539456564058, 3.80559388466228488940765137014, 3.82051151121896728239863977875, 4.26250676419046964951487041762, 4.62564200572480354485948660772, 4.91146452031773010622554800347, 4.93343453075699766950193652077, 5.51443902568099902141323608903, 5.58384010389784811522242015628, 5.70298091042920946086717120353, 6.31335588900175068848086073253, 6.33605370284512729798904460401, 6.65695917651390382627637725198, 7.01275670838954465020833305254, 7.14965074264055476362853282340, 7.24486505499314037530679437588, 7.59480249378731988578623493736
