| L(s) = 1 | + 16·5-s − 6·9-s + 16·13-s − 8·17-s + 108·25-s + 80·29-s + 16·37-s − 72·41-s − 96·45-s + 180·49-s + 144·53-s − 176·61-s + 256·65-s + 40·73-s + 27·81-s − 128·85-s + 136·89-s − 264·97-s + 208·101-s − 176·109-s + 328·113-s − 96·117-s + 132·121-s + 432·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 16/5·5-s − 2/3·9-s + 1.23·13-s − 0.470·17-s + 4.31·25-s + 2.75·29-s + 0.432·37-s − 1.75·41-s − 2.13·45-s + 3.67·49-s + 2.71·53-s − 2.88·61-s + 3.93·65-s + 0.547·73-s + 1/3·81-s − 1.50·85-s + 1.52·89-s − 2.72·97-s + 2.05·101-s − 1.61·109-s + 2.90·113-s − 0.820·117-s + 1.09·121-s + 3.45·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.03138392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.03138392\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 p T^{2} + 9062 T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 258 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 198 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1092 T^{2} + 534182 T^{4} - 1092 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 998 p^{2} T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 40 T + 1482 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1012 T^{2} + 112422 T^{4} - 1012 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 1218 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 36 T + 3302 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 900 T^{2} + 6425702 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2116 T^{2} + 7339782 T^{4} - 2116 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 72 T + 5738 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5762 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 88 T + 8994 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9156 T^{2} + 41992742 T^{4} - 9156 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 22417862 T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6516 T^{2} + 24592550 T^{4} - 6516 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13636 T^{2} + 114638502 T^{4} - 13636 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 68 T + 15462 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 132 T + 21638 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.04768040641855914320018436570, −6.76426855802276683061480539360, −6.75085265142214800656594094545, −6.43503861172733601644480021007, −6.22566617728812207218556579004, −5.81459409450596204527550246736, −5.79520011468008173507093568568, −5.76435413505731685460235336730, −5.61169481336504940163593873609, −5.01201121745033791497390879439, −4.98967070468349598792328807662, −4.67803537631472615827829871505, −4.32487101871259874572980145408, −4.07611535242803702264794815310, −3.75951139037406201394505404774, −3.35310772498921880676916585530, −3.01531973111013341368507533626, −2.80233275093591062667292651094, −2.47842774047467675162809256834, −2.23417411110678447708453411007, −1.94865925559611374108099866231, −1.66508507299746835591655952741, −1.28986370234061346779991869816, −0.74540668008873571396194302001, −0.64308671111890038849431412433,
0.64308671111890038849431412433, 0.74540668008873571396194302001, 1.28986370234061346779991869816, 1.66508507299746835591655952741, 1.94865925559611374108099866231, 2.23417411110678447708453411007, 2.47842774047467675162809256834, 2.80233275093591062667292651094, 3.01531973111013341368507533626, 3.35310772498921880676916585530, 3.75951139037406201394505404774, 4.07611535242803702264794815310, 4.32487101871259874572980145408, 4.67803537631472615827829871505, 4.98967070468349598792328807662, 5.01201121745033791497390879439, 5.61169481336504940163593873609, 5.76435413505731685460235336730, 5.79520011468008173507093568568, 5.81459409450596204527550246736, 6.22566617728812207218556579004, 6.43503861172733601644480021007, 6.75085265142214800656594094545, 6.76426855802276683061480539360, 7.04768040641855914320018436570
