| L(s) = 1 | + 24·13-s + 4·25-s + 32·37-s − 2·49-s + 8·61-s + 8·73-s − 24·97-s + 16·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 308·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 6.65·13-s + 4/5·25-s + 5.26·37-s − 2/7·49-s + 1.02·61-s + 0.936·73-s − 2.43·97-s + 1.53·109-s − 3.63·121-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 + 23.6·169-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 + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.74064442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.74064442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−6.44061218308063451695588553613, −6.31009291638790608640779471824, −6.18444985092489853450887768189, −5.88712477898299920694646294413, −5.68506348261791168936360425480, −5.60541958274214311231796109574, −5.46164432859158609941928595229, −5.07077920023534064166516045123, −4.74583591199790861561420840714, −4.23133469647246136472888468125, −4.21013493065346535161220347645, −4.13559315672981738264155927278, −4.08696020160856823636743040497, −3.62456374164899762468618868548, −3.41191508142841726603185793721, −3.20985189463747322517125435449, −3.08563849860196136170921276318, −2.63647230669998500667089564230, −2.49392343193830673172627218675, −1.96641932298947941329521041450, −1.63789403645100692738610325000, −1.32360169918675921311023180693, −1.10941636903021293827344834732, −0.796277695346660624491462692128, −0.78316264427135727292073719821,
0.78316264427135727292073719821, 0.796277695346660624491462692128, 1.10941636903021293827344834732, 1.32360169918675921311023180693, 1.63789403645100692738610325000, 1.96641932298947941329521041450, 2.49392343193830673172627218675, 2.63647230669998500667089564230, 3.08563849860196136170921276318, 3.20985189463747322517125435449, 3.41191508142841726603185793721, 3.62456374164899762468618868548, 4.08696020160856823636743040497, 4.13559315672981738264155927278, 4.21013493065346535161220347645, 4.23133469647246136472888468125, 4.74583591199790861561420840714, 5.07077920023534064166516045123, 5.46164432859158609941928595229, 5.60541958274214311231796109574, 5.68506348261791168936360425480, 5.88712477898299920694646294413, 6.18444985092489853450887768189, 6.31009291638790608640779471824, 6.44061218308063451695588553613
