| L(s) = 1 | + 72·17-s + 68·25-s − 72·41-s + 100·49-s − 328·73-s − 504·89-s + 440·97-s + 504·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.23·17-s + 2.71·25-s − 1.75·41-s + 2.04·49-s − 4.49·73-s − 5.66·89-s + 4.53·97-s + 4.46·113-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.475207552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.475207552\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 430 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2446 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2258 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8546 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8354 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 3550 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{4} \) |
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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
−6.86422012190438359078361645025, −6.57644123522675135890136743314, −6.50491828713032080648408296047, −5.86257168586347795933530374307, −5.84985096964436186265942254322, −5.68859687183778914514713600335, −5.60964787291047453358608906415, −5.17344111271199071748647454401, −5.15432709463440302168375174477, −4.75023910016741785738829114648, −4.52605682725075321197939644386, −4.31167353552181212593726436827, −4.05892552590043108589947529837, −3.57673556573545803613706538956, −3.36012321745227300968423063325, −3.33760543740613650169015531026, −2.96244552248747365841215365970, −2.73650832318037320553074631291, −2.66505200497999106628863372588, −1.96490530206499529636561366592, −1.59950988926846922614105142926, −1.29978819046718036379502081392, −1.19489227647182569626389076944, −0.68575365698402557343219794074, −0.44067603246664855991975339588,
0.44067603246664855991975339588, 0.68575365698402557343219794074, 1.19489227647182569626389076944, 1.29978819046718036379502081392, 1.59950988926846922614105142926, 1.96490530206499529636561366592, 2.66505200497999106628863372588, 2.73650832318037320553074631291, 2.96244552248747365841215365970, 3.33760543740613650169015531026, 3.36012321745227300968423063325, 3.57673556573545803613706538956, 4.05892552590043108589947529837, 4.31167353552181212593726436827, 4.52605682725075321197939644386, 4.75023910016741785738829114648, 5.15432709463440302168375174477, 5.17344111271199071748647454401, 5.60964787291047453358608906415, 5.68859687183778914514713600335, 5.84985096964436186265942254322, 5.86257168586347795933530374307, 6.50491828713032080648408296047, 6.57644123522675135890136743314, 6.86422012190438359078361645025
