| L(s) = 1 | + 10·5-s − 34·7-s − 14·11-s + 6·17-s + 52·23-s − 9·25-s − 340·35-s − 34·43-s + 86·47-s + 555·49-s − 140·55-s − 70·61-s + 90·73-s + 476·77-s + 64·83-s + 60·85-s − 452·101-s + 520·115-s − 204·119-s − 333·121-s − 510·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·5-s − 4.85·7-s − 1.27·11-s + 6/17·17-s + 2.26·23-s − 0.359·25-s − 9.71·35-s − 0.790·43-s + 1.82·47-s + 11.3·49-s − 2.54·55-s − 1.14·61-s + 1.23·73-s + 6.18·77-s + 0.771·83-s + 0.705·85-s − 4.47·101-s + 4.52·115-s − 1.71·119-s − 2.75·121-s − 4.07·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8256886641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8256886641\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 + 26 p T^{2} + p^{4} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - p T + 42 T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 17 T + 156 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 240 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 488 T^{2} + 110958 T^{4} - 488 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 224 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 26 T + 1170 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2612 T^{2} + 3028998 T^{4} - 2612 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2888 T^{2} + 3926478 T^{4} - 2888 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 632 T^{2} + 1876206 T^{4} + 632 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5732 T^{2} + 13632006 T^{4} - 5732 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 17 T + 564 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 43 T + 3726 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4468 T^{2} + 13384518 T^{4} - 4468 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9092 T^{2} + 39064038 T^{4} - 9092 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 35 T + 7620 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7948 T^{2} + 50546310 T^{4} + 7948 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8548 T^{2} + 35470470 T^{4} - 8548 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 45 T + 11036 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16568 T^{2} + 134358510 T^{4} - 16568 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 32 T + 2862 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 7396 T^{2} + 86628486 T^{4} - 7396 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23588 T^{2} + 287906886 T^{4} - 23588 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.08836386444857457706480786143, −5.92789636946246042892829236897, −5.59685589848387407461644530126, −5.56790425098377351932582906927, −5.54339152044358087029621923722, −5.18825598884575102246588242179, −5.03486933194356508520482830013, −4.57319753666123868508719120765, −4.39425034042689366766813849446, −4.10094313275204776243367873170, −3.85834801496061482588278984212, −3.51270885425679627595765269591, −3.49660481014060633893714910031, −3.19198139028928648008705841987, −2.97846614283934586805442402227, −2.87409750681423334031966482645, −2.57558504083076969081856735867, −2.52093924931143569252709519568, −2.19284823676519757991527080399, −1.73777355508200041860713602127, −1.65572040906560433503066010716, −1.04285223906952864520779138649, −0.807235942050273117973538923413, −0.36159784590033115359459843080, −0.18282836705245276394543868731,
0.18282836705245276394543868731, 0.36159784590033115359459843080, 0.807235942050273117973538923413, 1.04285223906952864520779138649, 1.65572040906560433503066010716, 1.73777355508200041860713602127, 2.19284823676519757991527080399, 2.52093924931143569252709519568, 2.57558504083076969081856735867, 2.87409750681423334031966482645, 2.97846614283934586805442402227, 3.19198139028928648008705841987, 3.49660481014060633893714910031, 3.51270885425679627595765269591, 3.85834801496061482588278984212, 4.10094313275204776243367873170, 4.39425034042689366766813849446, 4.57319753666123868508719120765, 5.03486933194356508520482830013, 5.18825598884575102246588242179, 5.54339152044358087029621923722, 5.56790425098377351932582906927, 5.59685589848387407461644530126, 5.92789636946246042892829236897, 6.08836386444857457706480786143
