| L(s) = 1 | − 8·7-s + 12·13-s − 8·25-s + 32·31-s − 12·37-s + 32·49-s − 32·67-s + 28·73-s − 96·91-s + 12·97-s + 8·103-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 64·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 3.32·13-s − 8/5·25-s + 5.74·31-s − 1.97·37-s + 32/7·49-s − 3.90·67-s + 3.27·73-s − 10.0·91-s + 1.21·97-s + 0.788·103-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56753713705238076935838029161, −7.05977632213055431396055662264, −6.84015897009788873126815544347, −6.73399695271106497869699011571, −6.55562903589419399218114980432, −6.15554584833467913489118584467, −6.12559248645424503146381628040, −6.03178310616356836964693964274, −5.79413736096819793170551158263, −5.69862461239992987827074594328, −4.97099835435441550210500193027, −4.80131505047026589461807564105, −4.54940342328145707829952932644, −4.16675319724783795628503102288, −3.94579918076548362111975035057, −3.74495328496290865116212101767, −3.33096957499814993653475337940, −3.13488813947697232335197629858, −3.08606963579875404429779005416, −2.86545373662246088416561761598, −2.17944852982505820453673977806, −1.91820950984907582639690913729, −1.34648498631240175762356989187, −0.77266078706127326832727693774, −0.59829248653955591051148185110,
0.59829248653955591051148185110, 0.77266078706127326832727693774, 1.34648498631240175762356989187, 1.91820950984907582639690913729, 2.17944852982505820453673977806, 2.86545373662246088416561761598, 3.08606963579875404429779005416, 3.13488813947697232335197629858, 3.33096957499814993653475337940, 3.74495328496290865116212101767, 3.94579918076548362111975035057, 4.16675319724783795628503102288, 4.54940342328145707829952932644, 4.80131505047026589461807564105, 4.97099835435441550210500193027, 5.69862461239992987827074594328, 5.79413736096819793170551158263, 6.03178310616356836964693964274, 6.12559248645424503146381628040, 6.15554584833467913489118584467, 6.55562903589419399218114980432, 6.73399695271106497869699011571, 6.84015897009788873126815544347, 7.05977632213055431396055662264, 7.56753713705238076935838029161
