| L(s) = 1 | − 6·7-s + 14·9-s − 6·19-s + 56·31-s − 106·41-s − 100·47-s − 129·49-s + 2·59-s − 84·63-s − 256·67-s − 14·71-s + 74·81-s − 158·97-s − 278·101-s + 374·103-s + 422·107-s + 194·109-s + 242·113-s + 222·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 6/7·7-s + 14/9·9-s − 0.315·19-s + 1.80·31-s − 2.58·41-s − 2.12·47-s − 2.63·49-s + 2/59·59-s − 4/3·63-s − 3.82·67-s − 0.197·71-s + 0.913·81-s − 1.62·97-s − 2.75·101-s + 3.63·103-s + 3.94·107-s + 1.77·109-s + 2.14·113-s + 1.83·121-s + 0.00787·127-s + 0.00763·131-s + 0.270·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4607492738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4607492738\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 31 | $D_{4}$ | \( 1 - 56 T + 2350 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 14 T^{2} + 122 T^{4} - 14 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 3 T + 78 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 222 T^{2} + 37242 T^{4} - 222 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 62 T^{2} + 53722 T^{4} - 62 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 300 T^{2} + 183846 T^{4} + 300 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 3 T + 524 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1532 T^{2} + 1076662 T^{4} - 1532 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 686 T^{2} + 1506490 T^{4} - 686 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4270 T^{2} + 7953306 T^{4} - 4270 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 53 T + 2974 T^{2} + 53 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 7374 T^{2} + 20431482 T^{4} - 7374 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 50 T + 2818 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 5150 T^{2} + 18220666 T^{4} - 5150 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - T + 5160 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 5646 T^{2} + 29782650 T^{4} - 5646 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 128 T + 12718 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 7 T + 5088 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 18924 T^{2} + 144957462 T^{4} - 18924 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 2036 T^{2} - 50938010 T^{4} - 2036 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 9134 T^{2} + 113467162 T^{4} - 9134 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 8892 T^{2} + 142105782 T^{4} - 8892 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 79 T + 16618 T^{2} + 79 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.91485057944114054571503691747, −5.90655335973125058329821566757, −5.74203965099805458521420823973, −5.23142833704441453683803940092, −5.08822220838643116917665229356, −4.83362941386775062717424335708, −4.66436897902440226576328703833, −4.61506741369509424782618236844, −4.54098769996464262729090019141, −4.11179117277469727284679945752, −3.89457965842028280496487118782, −3.55891252559388535300084234886, −3.51840389244490458391918860771, −3.07819686087981391282624308591, −3.07084762864685770036388290142, −2.94387508280933525524838859313, −2.64472491514558632511403464700, −2.01336932736917778632982986801, −1.93143994174596908174003825514, −1.68000379848997505611717116638, −1.62531447204519033187070335011, −1.17524434403510856190356950872, −0.891076265340757477392414882284, −0.47126140340091443299099781471, −0.088078225725203929812541497953,
0.088078225725203929812541497953, 0.47126140340091443299099781471, 0.891076265340757477392414882284, 1.17524434403510856190356950872, 1.62531447204519033187070335011, 1.68000379848997505611717116638, 1.93143994174596908174003825514, 2.01336932736917778632982986801, 2.64472491514558632511403464700, 2.94387508280933525524838859313, 3.07084762864685770036388290142, 3.07819686087981391282624308591, 3.51840389244490458391918860771, 3.55891252559388535300084234886, 3.89457965842028280496487118782, 4.11179117277469727284679945752, 4.54098769996464262729090019141, 4.61506741369509424782618236844, 4.66436897902440226576328703833, 4.83362941386775062717424335708, 5.08822220838643116917665229356, 5.23142833704441453683803940092, 5.74203965099805458521420823973, 5.90655335973125058329821566757, 5.91485057944114054571503691747
