| L(s) = 1 | − 2·5-s − 6·13-s − 2·17-s − 25-s − 10·29-s − 16·37-s − 4·41-s − 49-s − 16·53-s − 26·61-s + 12·65-s + 30·73-s + 4·85-s − 28·89-s + 36·97-s − 26·101-s − 24·109-s − 58·113-s − 8·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.66·13-s − 0.485·17-s − 1/5·25-s − 1.85·29-s − 2.63·37-s − 0.624·41-s − 1/7·49-s − 2.19·53-s − 3.32·61-s + 1.48·65-s + 3.51·73-s + 0.433·85-s − 2.96·89-s + 3.65·97-s − 2.58·101-s − 2.29·109-s − 5.45·113-s − 0.727·121-s − 0.178·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + T^{2} + 24 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} - 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 31 T^{2} + 888 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 65 T^{2} + 2040 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 5 T + 56 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 11 T^{2} + 1284 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 136 T^{2} + 8025 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 161 T^{2} + 10824 T^{4} + 161 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 56 T^{2} + 7449 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 88 T^{2} + 10617 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 176 T^{2} + 16638 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 15 T + 194 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 181 T^{2} + 20004 T^{4} + 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 125 T^{2} + 8700 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 14 T + 194 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
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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
−6.39604447969081742607165512599, −5.87763863669972079811034004703, −5.72732758305758865164229543020, −5.49262657295991354840521866469, −5.41919480121882032837815467096, −5.10668172839968728438619596676, −4.91739740392318554406929996570, −4.87253432852668878120270537488, −4.83990519440724490500025583897, −4.41599196432482173680492408828, −4.13123998994128293392684439472, −3.94677007308929328465308995655, −3.83285657836590094750976037086, −3.73121090632161564719538564471, −3.36805096861447066646272745118, −3.26829034825302020779690418568, −2.93478668222142794957258329292, −2.80576497505157332918829586907, −2.39741627739090106360538873672, −2.24946358160838918555320908673, −2.22187801125940561999341022673, −1.57159963207649862112085293038, −1.55588133669801280339149190051, −1.37574407165362149237193546842, −1.00757250932946861833644876816, 0, 0, 0, 0,
1.00757250932946861833644876816, 1.37574407165362149237193546842, 1.55588133669801280339149190051, 1.57159963207649862112085293038, 2.22187801125940561999341022673, 2.24946358160838918555320908673, 2.39741627739090106360538873672, 2.80576497505157332918829586907, 2.93478668222142794957258329292, 3.26829034825302020779690418568, 3.36805096861447066646272745118, 3.73121090632161564719538564471, 3.83285657836590094750976037086, 3.94677007308929328465308995655, 4.13123998994128293392684439472, 4.41599196432482173680492408828, 4.83990519440724490500025583897, 4.87253432852668878120270537488, 4.91739740392318554406929996570, 5.10668172839968728438619596676, 5.41919480121882032837815467096, 5.49262657295991354840521866469, 5.72732758305758865164229543020, 5.87763863669972079811034004703, 6.39604447969081742607165512599
