| L(s) = 1 | − 12·13-s + 32·31-s − 28·37-s + 16·43-s − 48·61-s − 32·67-s + 12·73-s − 20·97-s + 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.32·13-s + 5.74·31-s − 4.60·37-s + 2.43·43-s − 6.14·61-s − 3.90·67-s + 1.40·73-s − 2.03·97-s + 1.57·103-s − 1.81·121-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 + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3975835001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3975835001\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 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 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−6.04668532543660502527654888457, −5.97440971680816426545138835439, −5.56252525130084574154168737374, −5.55400025628583316592498623798, −5.17285351006806805083530414906, −4.89273061690963016493473311421, −4.76641505781645766909473015919, −4.56740093947293150311505605438, −4.52014291113320582051759004203, −4.40064059063919626550316417830, −4.34712062874239944329647649561, −3.67943489776109016588564801614, −3.55662933307974575407777360332, −3.17360639237138299437903135496, −2.93211582289602428292844811327, −2.90866945676150514639396811472, −2.82254397250739919130189069978, −2.37740877097733634816362643243, −2.29089405216119942479353105434, −1.80322266465904047654941160944, −1.71838982383836862301348600696, −1.30288202343338596925516312867, −1.04013909879030420156710336846, −0.51720286077981030061341818356, −0.12056875026187638384639040697,
0.12056875026187638384639040697, 0.51720286077981030061341818356, 1.04013909879030420156710336846, 1.30288202343338596925516312867, 1.71838982383836862301348600696, 1.80322266465904047654941160944, 2.29089405216119942479353105434, 2.37740877097733634816362643243, 2.82254397250739919130189069978, 2.90866945676150514639396811472, 2.93211582289602428292844811327, 3.17360639237138299437903135496, 3.55662933307974575407777360332, 3.67943489776109016588564801614, 4.34712062874239944329647649561, 4.40064059063919626550316417830, 4.52014291113320582051759004203, 4.56740093947293150311505605438, 4.76641505781645766909473015919, 4.89273061690963016493473311421, 5.17285351006806805083530414906, 5.55400025628583316592498623798, 5.56252525130084574154168737374, 5.97440971680816426545138835439, 6.04668532543660502527654888457
