| L(s) = 1 | + 6·9-s + 56·17-s + 4·25-s − 56·41-s − 92·49-s − 200·73-s + 27·81-s + 248·89-s − 584·97-s + 520·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 336·153-s + 157-s + 163-s + 167-s + 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 3.29·17-s + 4/25·25-s − 1.36·41-s − 1.87·49-s − 2.73·73-s + 1/3·81-s + 2.78·89-s − 6.02·97-s + 4.60·113-s − 3.20·121-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 + 2.19·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.360060021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360060021\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3650 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 766 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4610 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9506 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{4} \) |
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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
−7.87300871015579292895366216876, −7.81047130018142817851123142398, −7.65877302346149446504204491785, −7.30997158540891970705700473266, −6.92490507681346613242178095855, −6.77952368631480579094103210185, −6.58149189688274469489882165739, −6.13785766961229702628623374441, −5.95234175785605594309805176434, −5.69789424295214023214015001789, −5.25072535788889615179863866723, −5.24723097026686343974978511357, −4.98472165906104631554677850699, −4.56174206951935703693893312639, −4.20552734000691682364851634956, −3.99420422629360208134696246011, −3.48388600920688756217473604560, −3.32618197500386366211747823630, −3.16920648551880259251113855575, −2.66824947096284882253026439140, −2.32246255945711719675453024730, −1.58477370440563507799928079507, −1.34286285132347209182417282391, −1.21046312263339819291986840948, −0.32390131585895781729059580845,
0.32390131585895781729059580845, 1.21046312263339819291986840948, 1.34286285132347209182417282391, 1.58477370440563507799928079507, 2.32246255945711719675453024730, 2.66824947096284882253026439140, 3.16920648551880259251113855575, 3.32618197500386366211747823630, 3.48388600920688756217473604560, 3.99420422629360208134696246011, 4.20552734000691682364851634956, 4.56174206951935703693893312639, 4.98472165906104631554677850699, 5.24723097026686343974978511357, 5.25072535788889615179863866723, 5.69789424295214023214015001789, 5.95234175785605594309805176434, 6.13785766961229702628623374441, 6.58149189688274469489882165739, 6.77952368631480579094103210185, 6.92490507681346613242178095855, 7.30997158540891970705700473266, 7.65877302346149446504204491785, 7.81047130018142817851123142398, 7.87300871015579292895366216876
