| L(s) = 1 | − 4·4-s + 4·16-s − 16·19-s + 40·25-s + 8·49-s + 16·64-s − 96·73-s + 64·76-s − 160·100-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 32·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 16-s − 3.67·19-s + 8·25-s + 8/7·49-s + 2·64-s − 11.2·73-s + 7.34·76-s − 16·100-s − 4.36·121-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 − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2.28·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1291652248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1291652248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 19 | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - p T^{2} )^{8} \) |
| 7 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.15967133074733944696950561325, −4.11614268779359623289803513671, −3.79465213999867927558712885276, −3.71483568082702740303754892285, −3.59906498643372312398371076186, −3.58638793025680342308080995334, −3.16082571277051090695649680488, −3.07611112779437945429674353468, −2.98896350510321427867348845745, −2.98443064598103847496595649296, −2.77176384375989408696164551248, −2.58655123966279304473635688526, −2.52457442625581565413530075563, −2.49502210378172303780187543777, −2.37578242542297392651074979599, −1.86995344646743254585171739210, −1.75852462265135540497199257052, −1.72146450280527091679784151152, −1.41349188524016326130338442361, −1.26942789262007981454662609880, −1.04223324214299789113984332669, −0.995516195806499926593447299045, −0.69985080228520100585745762067, −0.33605209465174742598442123407, −0.06695673624311332273127829148,
0.06695673624311332273127829148, 0.33605209465174742598442123407, 0.69985080228520100585745762067, 0.995516195806499926593447299045, 1.04223324214299789113984332669, 1.26942789262007981454662609880, 1.41349188524016326130338442361, 1.72146450280527091679784151152, 1.75852462265135540497199257052, 1.86995344646743254585171739210, 2.37578242542297392651074979599, 2.49502210378172303780187543777, 2.52457442625581565413530075563, 2.58655123966279304473635688526, 2.77176384375989408696164551248, 2.98443064598103847496595649296, 2.98896350510321427867348845745, 3.07611112779437945429674353468, 3.16082571277051090695649680488, 3.58638793025680342308080995334, 3.59906498643372312398371076186, 3.71483568082702740303754892285, 3.79465213999867927558712885276, 4.11614268779359623289803513671, 4.15967133074733944696950561325
Plot not available for L-functions of degree greater than 10.
