| L(s) = 1 | + 2-s + 4-s + 2·7-s + 3·8-s + 2·14-s + 16-s + 12·19-s + 6·25-s + 2·28-s + 8·29-s − 32-s − 12·37-s + 12·38-s + 8·47-s + 6·49-s + 6·50-s − 16·53-s + 6·56-s + 8·58-s − 16·59-s + 64-s − 12·74-s + 12·76-s + 8·83-s + 8·94-s + 6·98-s + 6·100-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 1.06·8-s + 0.534·14-s + 1/4·16-s + 2.75·19-s + 6/5·25-s + 0.377·28-s + 1.48·29-s − 0.176·32-s − 1.97·37-s + 1.94·38-s + 1.16·47-s + 6/7·49-s + 0.848·50-s − 2.19·53-s + 0.801·56-s + 1.05·58-s − 2.08·59-s + 1/8·64-s − 1.39·74-s + 1.37·76-s + 0.878·83-s + 0.825·94-s + 0.606·98-s + 3/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.775611140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.775611140\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.882830805806560169550438213732, −8.489556537432565556908423767461, −8.096208283443308791556499903929, −8.013416670888288132814655353767, −7.72247501793234012286361661719, −7.31965919028364277186617675124, −7.24034433777827634857816587182, −7.14157498086366932121834693991, −6.45480972987674849815622195444, −6.37462802900158648085082059336, −6.36679677673830833547382412002, −5.50455219882482109082299537435, −5.39791744168898827450635389077, −5.18723591545151194904594203404, −5.05811489511074808848571505015, −4.56737414060814760194025554038, −4.40397294426286718002719676220, −4.02922784317045736189504813013, −3.54650692368533166345663294916, −3.23509725083278279780199678671, −2.81531726953070098416216895727, −2.70433685782308058967524019006, −1.86129785118577119109976243663, −1.48609842293146197234493075877, −1.06011230503244344915542757204,
1.06011230503244344915542757204, 1.48609842293146197234493075877, 1.86129785118577119109976243663, 2.70433685782308058967524019006, 2.81531726953070098416216895727, 3.23509725083278279780199678671, 3.54650692368533166345663294916, 4.02922784317045736189504813013, 4.40397294426286718002719676220, 4.56737414060814760194025554038, 5.05811489511074808848571505015, 5.18723591545151194904594203404, 5.39791744168898827450635389077, 5.50455219882482109082299537435, 6.36679677673830833547382412002, 6.37462802900158648085082059336, 6.45480972987674849815622195444, 7.14157498086366932121834693991, 7.24034433777827634857816587182, 7.31965919028364277186617675124, 7.72247501793234012286361661719, 8.013416670888288132814655353767, 8.096208283443308791556499903929, 8.489556537432565556908423767461, 8.882830805806560169550438213732
