| L(s) = 1 | + 6·9-s + 32·11-s + 8·17-s − 32·19-s + 40·41-s + 32·43-s + 76·49-s + 128·59-s − 256·67-s − 200·73-s + 27·81-s + 160·83-s − 200·89-s − 56·97-s + 192·99-s − 320·107-s + 344·113-s + 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.90·11-s + 8/17·17-s − 1.68·19-s + 0.975·41-s + 0.744·43-s + 1.55·49-s + 2.16·59-s − 3.82·67-s − 2.73·73-s + 1/3·81-s + 1.92·83-s − 2.24·89-s − 0.577·97-s + 1.93·99-s − 2.99·107-s + 3.04·113-s + 1.28·121-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 + 0.313·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.356106551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356106551\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 390 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 738 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 64 T + 7554 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 100 T + 10086 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 80 T + 14610 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 28 T + 12102 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.30837432617801267022212997253, −5.99852341962762696348607485779, −5.97453157257984598381081181868, −5.43038858108407311699709910724, −5.40154869760803525790748042018, −5.16790227980571358819034513640, −5.02697131928056970415322728439, −4.36640152925299006331163329614, −4.28995315262312354728827752575, −4.27681553971705068375822292030, −4.14201753226866515679781318300, −3.99386139666729341941121198164, −3.71084394370455828260012367088, −3.40557016520608517243966409130, −3.12864895466725836463511195270, −2.90004736955513344832554781072, −2.46382369906127906732552316927, −2.29250095185819982617812541981, −2.27909898829777739531796402702, −1.46692307426145807498682876334, −1.45555949584384425579074327990, −1.36812250922288461757431683009, −1.10424320769343015670376940787, −0.60679995983731705730875909482, −0.16706220649453262561616381310,
0.16706220649453262561616381310, 0.60679995983731705730875909482, 1.10424320769343015670376940787, 1.36812250922288461757431683009, 1.45555949584384425579074327990, 1.46692307426145807498682876334, 2.27909898829777739531796402702, 2.29250095185819982617812541981, 2.46382369906127906732552316927, 2.90004736955513344832554781072, 3.12864895466725836463511195270, 3.40557016520608517243966409130, 3.71084394370455828260012367088, 3.99386139666729341941121198164, 4.14201753226866515679781318300, 4.27681553971705068375822292030, 4.28995315262312354728827752575, 4.36640152925299006331163329614, 5.02697131928056970415322728439, 5.16790227980571358819034513640, 5.40154869760803525790748042018, 5.43038858108407311699709910724, 5.97453157257984598381081181868, 5.99852341962762696348607485779, 6.30837432617801267022212997253
