| L(s) = 1 | − 2·5-s − 4·13-s − 14·17-s + 3·25-s − 8·29-s + 2·37-s + 12·41-s + 49-s − 20·53-s − 12·61-s + 8·65-s − 24·73-s + 28·85-s + 24·89-s − 16·97-s + 4·101-s − 30·109-s − 40·113-s − 16·121-s − 14·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.10·13-s − 3.39·17-s + 3/5·25-s − 1.48·29-s + 0.328·37-s + 1.87·41-s + 1/7·49-s − 2.74·53-s − 1.53·61-s + 0.992·65-s − 2.80·73-s + 3.03·85-s + 2.54·89-s − 1.62·97-s + 0.398·101-s − 2.87·109-s − 3.76·113-s − 1.45·121-s − 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 + T + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - T^{2} + 88 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 142 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 48 T^{2} + 1134 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 1366 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 71 T^{2} + 4456 T^{4} + 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 159 T^{2} + 10728 T^{4} + 159 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 48 T^{2} - 498 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 16 T^{2} + 8878 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 103 T^{2} + 12232 T^{4} + 103 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 4102 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 276 T^{2} + 32166 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.12029391731427315789513145521, −5.62180634999003842183532174611, −5.46833489669855354840731042818, −5.32362483713830418527763329447, −5.24047437668904746894271690896, −4.82108639931415739717508289450, −4.76524156985114923069895045056, −4.52179966955378123593433680420, −4.41581608334191897435239196048, −4.22847109885331744619884987749, −4.17230983459498159093735245994, −3.92214676061834151179740557016, −3.78185648876743735765045551287, −3.22923874769557115535923425146, −3.22828908652212673182644652733, −3.19257279859633192610989461106, −2.71226117087426344504330374653, −2.57667567965592075528854474191, −2.26389928203892770152407244227, −2.24788638689645544235796437667, −2.14373983451169288557061606311, −1.55717328523887547018131238917, −1.45264575210826105889514487468, −1.18695215174687380244086291652, −0.903903083101652986774411356564, 0, 0, 0, 0,
0.903903083101652986774411356564, 1.18695215174687380244086291652, 1.45264575210826105889514487468, 1.55717328523887547018131238917, 2.14373983451169288557061606311, 2.24788638689645544235796437667, 2.26389928203892770152407244227, 2.57667567965592075528854474191, 2.71226117087426344504330374653, 3.19257279859633192610989461106, 3.22828908652212673182644652733, 3.22923874769557115535923425146, 3.78185648876743735765045551287, 3.92214676061834151179740557016, 4.17230983459498159093735245994, 4.22847109885331744619884987749, 4.41581608334191897435239196048, 4.52179966955378123593433680420, 4.76524156985114923069895045056, 4.82108639931415739717508289450, 5.24047437668904746894271690896, 5.32362483713830418527763329447, 5.46833489669855354840731042818, 5.62180634999003842183532174611, 6.12029391731427315789513145521
