| L(s) = 1 | + 16·7-s − 8·13-s + 32·19-s − 10·25-s − 8·31-s + 136·37-s + 80·43-s + 144·49-s + 40·61-s − 304·67-s + 152·73-s − 200·79-s − 128·91-s − 424·97-s + 112·103-s − 104·109-s + 88·121-s + 127-s + 131-s + 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 0.615·13-s + 1.68·19-s − 2/5·25-s − 0.258·31-s + 3.67·37-s + 1.86·43-s + 2.93·49-s + 0.655·61-s − 4.53·67-s + 2.08·73-s − 2.53·79-s − 1.40·91-s − 4.37·97-s + 1.08·103-s − 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s + 3.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.325755961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325755961\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 8 T + 24 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 252 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 1566 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 68 T + 3084 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 3738 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 4302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 100 T + 11742 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 212 T + 29694 T^{2} + 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.91314475412389150952921207072, −5.88397787306025640520337871543, −5.65662858453707675383879941994, −5.23638906771885388915955998380, −5.22786533927295692713409910925, −5.14125440920176012543193124167, −4.81234169772479876157869346882, −4.46658913911784043237332720750, −4.36160006421020878300760201664, −4.26600508952183869611069059780, −4.10377800838640465868986811766, −3.89567027962258428786873764231, −3.49575094875659462259462389557, −3.06813700998091811170376018729, −3.03783036431720831154404217285, −2.67862726336953698913511559911, −2.50397431913309075524282700028, −2.44976854521958633413799941586, −1.84623490868083196124558802835, −1.80824935877262457188246710235, −1.43123494579099588369557609827, −1.06845278028112520599159101511, −1.05051438311073317238814767513, −0.75059489951598764493378784853, −0.10156633301517625104581771976,
0.10156633301517625104581771976, 0.75059489951598764493378784853, 1.05051438311073317238814767513, 1.06845278028112520599159101511, 1.43123494579099588369557609827, 1.80824935877262457188246710235, 1.84623490868083196124558802835, 2.44976854521958633413799941586, 2.50397431913309075524282700028, 2.67862726336953698913511559911, 3.03783036431720831154404217285, 3.06813700998091811170376018729, 3.49575094875659462259462389557, 3.89567027962258428786873764231, 4.10377800838640465868986811766, 4.26600508952183869611069059780, 4.36160006421020878300760201664, 4.46658913911784043237332720750, 4.81234169772479876157869346882, 5.14125440920176012543193124167, 5.22786533927295692713409910925, 5.23638906771885388915955998380, 5.65662858453707675383879941994, 5.88397787306025640520337871543, 5.91314475412389150952921207072
