| L(s) = 1 | − 32·5-s − 54·9-s − 480·13-s + 696·17-s + 156·25-s − 2.84e3·29-s + 672·37-s + 504·41-s + 1.72e3·45-s + 7.39e3·49-s − 160·53-s + 7.96e3·61-s + 1.53e4·65-s + 5.12e3·73-s + 2.18e3·81-s − 2.22e4·85-s + 1.58e4·89-s − 2.26e4·97-s + 3.01e4·101-s + 4.51e3·109-s − 120·113-s + 2.59e4·117-s + 1.31e4·121-s − 7.00e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.27·5-s − 2/3·9-s − 2.84·13-s + 2.40·17-s + 0.249·25-s − 3.38·29-s + 0.490·37-s + 0.299·41-s + 0.853·45-s + 3.08·49-s − 0.0569·53-s + 2.14·61-s + 3.63·65-s + 0.962·73-s + 1/3·81-s − 3.08·85-s + 1.99·89-s − 2.40·97-s + 2.95·101-s + 0.379·109-s − 0.00939·113-s + 1.89·117-s + 0.898·121-s − 0.448·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.825806358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825806358\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 + 16 T + 306 T^{2} + 16 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 7396 T^{2} + 24172614 T^{4} - 7396 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1196 p T^{2} + 368768646 T^{4} - 1196 p^{9} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 240 T + 67490 T^{2} + 240 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 348 T + 132806 T^{2} - 348 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 29668 T^{2} - 6278881530 T^{4} - 29668 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 293380 T^{2} + 44367843462 T^{4} - 293380 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1424 T + 2274 p^{2} T^{2} + 1424 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2304868 T^{2} + 2740651806150 T^{4} - 2304868 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 336 T + 3373346 T^{2} - 336 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 252 T + 441926 T^{2} - 252 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8877412 T^{2} + 38500496392326 T^{4} - 8877412 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2352772 T^{2} + 30062992642566 T^{4} - 2352772 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 80 T + 15152562 T^{2} + 80 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 22520114 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 3984 T + 23853794 T^{2} - 3984 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 64146148 T^{2} + 1777243586102790 T^{4} - 64146148 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6928132 T^{2} - 515199567726714 T^{4} - 6928132 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 2564 T + 57407814 T^{2} - 2564 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8064100 T^{2} + 1391113872282822 T^{4} - 8064100 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 127863268 T^{2} + 8102648542271238 T^{4} - 127863268 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 7908 T + 134667398 T^{2} - 7908 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 11300 T + 202529862 T^{2} + 11300 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−6.93846916300920843235975518288, −6.91102062257333395608421763179, −6.14989095518923267192738477291, −6.01353154909025883041488864519, −5.83173724546239487538726294409, −5.65598142231799267883429252022, −5.29850951394329622486168465520, −5.22801085913790014718541174938, −5.02173309814050758394465762178, −4.56312339651335715251407271771, −4.50324497334772381965886309264, −4.07758633361785718920620141267, −3.72815668390396722875002593965, −3.61449410902695006894465402974, −3.47654035542958521763835346256, −3.23062639887261087333972270436, −2.57052382315564330932474597316, −2.45956520375645605835045672648, −2.40551841403215961818942078671, −1.96311468022911079418904395135, −1.59953417915793974901434503938, −1.09365473435868774230321072666, −0.67663127283522962466601035877, −0.48731059735499076139972023921, −0.24714341642059371992173833528,
0.24714341642059371992173833528, 0.48731059735499076139972023921, 0.67663127283522962466601035877, 1.09365473435868774230321072666, 1.59953417915793974901434503938, 1.96311468022911079418904395135, 2.40551841403215961818942078671, 2.45956520375645605835045672648, 2.57052382315564330932474597316, 3.23062639887261087333972270436, 3.47654035542958521763835346256, 3.61449410902695006894465402974, 3.72815668390396722875002593965, 4.07758633361785718920620141267, 4.50324497334772381965886309264, 4.56312339651335715251407271771, 5.02173309814050758394465762178, 5.22801085913790014718541174938, 5.29850951394329622486168465520, 5.65598142231799267883429252022, 5.83173724546239487538726294409, 6.01353154909025883041488864519, 6.14989095518923267192738477291, 6.91102062257333395608421763179, 6.93846916300920843235975518288
