| L(s) = 1 | + 4·3-s + 10·9-s + 8·11-s + 8·17-s + 20·27-s + 32·33-s + 24·41-s − 16·43-s + 32·51-s + 16·59-s − 32·67-s + 16·73-s + 35·81-s + 24·83-s + 40·89-s + 24·97-s + 80·99-s − 16·107-s + 24·113-s + 12·121-s + 96·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 2.41·11-s + 1.94·17-s + 3.84·27-s + 5.57·33-s + 3.74·41-s − 2.43·43-s + 4.48·51-s + 2.08·59-s − 3.90·67-s + 1.87·73-s + 35/9·81-s + 2.63·83-s + 4.23·89-s + 2.43·97-s + 8.04·99-s − 1.54·107-s + 2.25·113-s + 1.09·121-s + 8.65·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.30214154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.30214154\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 18 T^{4} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 30 T^{4} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} + 582 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 1650 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 96 T^{2} + 4098 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 4342 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 6822 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 64 T^{2} + 1234 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 15766 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 172 T^{2} + 15430 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 288 T^{2} + 33090 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.64021612965803747033709659373, −6.54329110643883996842409073755, −6.31224745666468120346977912067, −6.29288973439363982951296146147, −6.06110014603452595243852196163, −5.64346013649349542096331190292, −5.43271345024168712166410217227, −5.16258685120708637575891017186, −4.84740150890825885802562223034, −4.66617330088337856098167598277, −4.40809179445598129659432166342, −4.19104186743646191051804081604, −3.94680370234418671296083560323, −3.53839627562226145909601721397, −3.49119461920987993867327191351, −3.43610270938515751061278790766, −3.29058823319757676057720587674, −2.70764034926847982558398302405, −2.49457424504648060187593979008, −2.14215966123727427823337507029, −2.12180394044258468573790142845, −1.44835202152548496308962733591, −1.42665027739424900476450842867, −0.901332249465280244979342675905, −0.800591001250872551317507586273,
0.800591001250872551317507586273, 0.901332249465280244979342675905, 1.42665027739424900476450842867, 1.44835202152548496308962733591, 2.12180394044258468573790142845, 2.14215966123727427823337507029, 2.49457424504648060187593979008, 2.70764034926847982558398302405, 3.29058823319757676057720587674, 3.43610270938515751061278790766, 3.49119461920987993867327191351, 3.53839627562226145909601721397, 3.94680370234418671296083560323, 4.19104186743646191051804081604, 4.40809179445598129659432166342, 4.66617330088337856098167598277, 4.84740150890825885802562223034, 5.16258685120708637575891017186, 5.43271345024168712166410217227, 5.64346013649349542096331190292, 6.06110014603452595243852196163, 6.29288973439363982951296146147, 6.31224745666468120346977912067, 6.54329110643883996842409073755, 6.64021612965803747033709659373
