| L(s) = 1 | + 4·7-s − 16·11-s + 32·13-s + 40·17-s + 56·23-s + 16·31-s − 64·37-s − 56·41-s − 8·43-s + 128·47-s + 8·49-s − 56·53-s + 200·61-s − 200·67-s + 272·71-s − 76·73-s − 64·77-s − 9·81-s − 16·83-s + 128·91-s + 20·97-s + 136·101-s − 308·103-s + 128·107-s − 224·113-s + 160·119-s − 216·121-s + ⋯ |
| L(s) = 1 | + 4/7·7-s − 1.45·11-s + 2.46·13-s + 2.35·17-s + 2.43·23-s + 0.516·31-s − 1.72·37-s − 1.36·41-s − 0.186·43-s + 2.72·47-s + 8/49·49-s − 1.05·53-s + 3.27·61-s − 2.98·67-s + 3.83·71-s − 1.04·73-s − 0.831·77-s − 1/9·81-s − 0.192·83-s + 1.40·91-s + 0.206·97-s + 1.34·101-s − 2.99·103-s + 1.19·107-s − 1.98·113-s + 1.34·119-s − 1.78·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.021077913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.021077913\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 156 T^{3} + 2942 T^{4} - 156 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 204 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 9120 T^{3} + 148994 T^{4} - 9120 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 15240 T^{3} + 281858 T^{4} - 15240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 50904 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1722 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T + 2048 T^{2} + 58176 T^{3} + 1440962 T^{4} + 58176 p^{2} T^{5} + 2048 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 5256 T^{3} - 557566 T^{4} + 5256 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 506496 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 155064 T^{3} + 15333122 T^{4} + 155064 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 200 T + 20000 T^{2} + 1888200 T^{3} + 153742658 T^{4} + 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} + 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 76 T + 2888 T^{2} - 65436 T^{3} - 36833458 T^{4} - 65436 p^{2} T^{5} + 2888 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 101328 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 173820 T^{3} + 150551438 T^{4} - 173820 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.88785304801776274094565408926, −6.51059468600811607625793337532, −6.26468829056125915631962076391, −6.15392577428109212029280803199, −5.82050983149235629695174848480, −5.44745642748469215192040241208, −5.41103080061491809403738222687, −5.20886421391440096425202650678, −5.20420649178663143973088203296, −4.87627907550082784747357542811, −4.49302057937257021683009906832, −4.17767079679655014117329959557, −3.89142086321102563736319360407, −3.65030115971014969694633377534, −3.55599230790551428682359355946, −3.20729464741117359727453362901, −2.96520432291598868154298734446, −2.63578999032072642175749041473, −2.60707029133905077735632970980, −1.89603427580325878357416799726, −1.70338471001732256700438754124, −1.24332589706096673458788854861, −1.18184840826946378040987695054, −0.68458432275619475398891340959, −0.47876994088469545061825708378,
0.47876994088469545061825708378, 0.68458432275619475398891340959, 1.18184840826946378040987695054, 1.24332589706096673458788854861, 1.70338471001732256700438754124, 1.89603427580325878357416799726, 2.60707029133905077735632970980, 2.63578999032072642175749041473, 2.96520432291598868154298734446, 3.20729464741117359727453362901, 3.55599230790551428682359355946, 3.65030115971014969694633377534, 3.89142086321102563736319360407, 4.17767079679655014117329959557, 4.49302057937257021683009906832, 4.87627907550082784747357542811, 5.20420649178663143973088203296, 5.20886421391440096425202650678, 5.41103080061491809403738222687, 5.44745642748469215192040241208, 5.82050983149235629695174848480, 6.15392577428109212029280803199, 6.26468829056125915631962076391, 6.51059468600811607625793337532, 6.88785304801776274094565408926
