| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 16·6-s + 4·7-s − 20·8-s + 8·9-s − 12·11-s − 40·12-s − 16·14-s + 35·16-s − 32·18-s − 16·21-s + 48·22-s + 80·24-s + 8·25-s − 16·27-s + 40·28-s + 4·31-s − 56·32-s + 48·33-s + 80·36-s + 4·37-s + 12·41-s + 64·42-s + 32·43-s − 120·44-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 8/3·9-s − 3.61·11-s − 11.5·12-s − 4.27·14-s + 35/4·16-s − 7.54·18-s − 3.49·21-s + 10.2·22-s + 16.3·24-s + 8/5·25-s − 3.07·27-s + 7.55·28-s + 0.718·31-s − 9.89·32-s + 8.35·33-s + 40/3·36-s + 0.657·37-s + 1.87·41-s + 9.87·42-s + 4.87·43-s − 18.0·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1446367425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1446367425\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 14 T^{2} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1567 T^{4} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 192 T^{3} - 1633 T^{4} + 192 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 1138 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5854 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 166 T^{2} + 11235 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 3555 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1344 T^{3} + 13583 T^{4} - 1344 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 52 T^{2} - 714 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 1056 T^{3} + 15199 T^{4} - 1056 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - p^{2} T^{4} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 11234 T^{4} - 1200 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 768 T^{3} + 18431 T^{4} + 768 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.331093417074770181909538288357, −9.246356378994868481546662565321, −8.942457384709014871111188449552, −8.391024197250654595612922122257, −8.120643014767994056446243577361, −8.072766323650021338088375161053, −7.82142094802301665694352217739, −7.46490786699537598006106925081, −7.37616267337189092173334361843, −7.20715703613127869885283793824, −6.57809881474586779126552393297, −6.36677264876718657172552764850, −5.98718256841431981560261418285, −5.53819182592103533101144480939, −5.53805942215431286644798672492, −5.42362586164330365067503332566, −4.74485787140555317035328819439, −4.61396679101289081291319229855, −4.15558386491829335196254157590, −3.18588544431618417814849432297, −2.69736508549986738926955025885, −2.27594636417031647619080348140, −2.12881836399994994960256539840, −0.876178125641113829981671910039, −0.70177386702254961580331789737,
0.70177386702254961580331789737, 0.876178125641113829981671910039, 2.12881836399994994960256539840, 2.27594636417031647619080348140, 2.69736508549986738926955025885, 3.18588544431618417814849432297, 4.15558386491829335196254157590, 4.61396679101289081291319229855, 4.74485787140555317035328819439, 5.42362586164330365067503332566, 5.53805942215431286644798672492, 5.53819182592103533101144480939, 5.98718256841431981560261418285, 6.36677264876718657172552764850, 6.57809881474586779126552393297, 7.20715703613127869885283793824, 7.37616267337189092173334361843, 7.46490786699537598006106925081, 7.82142094802301665694352217739, 8.072766323650021338088375161053, 8.120643014767994056446243577361, 8.391024197250654595612922122257, 8.942457384709014871111188449552, 9.246356378994868481546662565321, 9.331093417074770181909538288357
