L(s) = 1 | + 4-s − 6·9-s − 3·16-s − 16·19-s − 6·36-s + 28·49-s − 7·64-s − 16·76-s + 27·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 18·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 96·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 2·9-s − 3/4·16-s − 3.67·19-s − 36-s + 4·49-s − 7/8·64-s − 1.83·76-s + 3·81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 7.34·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9302514048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9302514048\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−7.66641937284352285278271701363, −7.41583346867162781325024576245, −7.20783895120409382744131406582, −6.85808970779806917810203595857, −6.75215948023137960166942279914, −6.52407905451430287292762328100, −6.24568130251168253340295916144, −5.89129961805469230636625820507, −5.85457304891056004505856214784, −5.56007435356991714005139779570, −5.54770102164426611730704694018, −4.78015580904738599894437729781, −4.75978776252473439105579639596, −4.43884623727815658805114866687, −4.26665633937315298954607976066, −3.77864372974204266585020853295, −3.71172515774603991593557033322, −3.23855542968127472546251401678, −2.85492485381580813332382201670, −2.54206740467330219094156722936, −2.34633034388162278141777002583, −1.98794904897675512345100836724, −1.91784268102805114717027825153, −0.891855074449704594312814741654, −0.32440844774353554180759907523,
0.32440844774353554180759907523, 0.891855074449704594312814741654, 1.91784268102805114717027825153, 1.98794904897675512345100836724, 2.34633034388162278141777002583, 2.54206740467330219094156722936, 2.85492485381580813332382201670, 3.23855542968127472546251401678, 3.71172515774603991593557033322, 3.77864372974204266585020853295, 4.26665633937315298954607976066, 4.43884623727815658805114866687, 4.75978776252473439105579639596, 4.78015580904738599894437729781, 5.54770102164426611730704694018, 5.56007435356991714005139779570, 5.85457304891056004505856214784, 5.89129961805469230636625820507, 6.24568130251168253340295916144, 6.52407905451430287292762328100, 6.75215948023137960166942279914, 6.85808970779806917810203595857, 7.20783895120409382744131406582, 7.41583346867162781325024576245, 7.66641937284352285278271701363