| L(s) = 1 | − 4·2-s + 2·3-s + 8·4-s − 8·6-s − 2·7-s − 8·8-s + 9-s + 16·12-s + 8·14-s − 4·16-s + 24·17-s − 4·18-s + 2·19-s − 4·21-s − 16·24-s + 25-s − 2·27-s − 16·28-s + 24·29-s + 6·31-s + 32·32-s − 96·34-s + 8·36-s + 2·37-s − 8·38-s + 16·42-s − 8·47-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 4·4-s − 3.26·6-s − 0.755·7-s − 2.82·8-s + 1/3·9-s + 4.61·12-s + 2.13·14-s − 16-s + 5.82·17-s − 0.942·18-s + 0.458·19-s − 0.872·21-s − 3.26·24-s + 1/5·25-s − 0.384·27-s − 3.02·28-s + 4.45·29-s + 1.07·31-s + 5.65·32-s − 16.4·34-s + 4/3·36-s + 0.328·37-s − 1.29·38-s + 2.46·42-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9891023320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9891023320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 651 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 22 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 18 T^{3} + 1404 T^{4} + 18 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 59 T^{2} + 22 T^{3} + 2452 T^{4} + 22 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 3819 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 34 T^{2} + 32 T^{3} + 4387 T^{4} + 32 p T^{5} - 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 14 T^{2} + 288 T^{3} + 6459 T^{4} + 288 p T^{5} + 14 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T - 22 T^{2} + 256 T^{3} + 139 T^{4} + 256 p T^{5} - 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 24 T + 346 T^{2} + 3696 T^{3} + 31707 T^{4} + 3696 p T^{5} + 346 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 253 T^{2} - 2610 T^{3} + 23772 T^{4} - 2610 p T^{5} + 253 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 3366 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 3042 p T^{5} + 277 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T + 137 T^{2} + 750 T^{3} + 10332 T^{4} + 750 p T^{5} + 137 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 36 T + 682 T^{2} + 9000 T^{3} + 93027 T^{4} + 9000 p T^{5} + 682 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13254 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.097736411986031471853808065656, −7.999341954034409427258602619972, −7.85122310648986188901733607922, −7.70316411877931333343373067411, −7.56567951674639029647512804182, −6.80822835808636887413494411107, −6.76530102180337152292914179157, −6.64009052540507893477019120956, −6.52407394181042980603428275582, −5.92073153062059669166155810561, −5.66232878047300777304000345770, −5.33331584176933187633913425429, −5.12531825747921367986760931213, −4.75911317266918689746839673193, −4.47458405778873887908822078185, −3.88682653668595991178059538036, −3.76394954278801020592040070017, −3.01810365918809937839150727925, −2.95882902574004375627074912277, −2.89111900236793762470330141692, −2.74907569056386376195936519070, −1.62911439397576681304264936851, −1.32767783393785903317366802082, −1.22525235011724995360000496144, −0.69344408710602790532978639240,
0.69344408710602790532978639240, 1.22525235011724995360000496144, 1.32767783393785903317366802082, 1.62911439397576681304264936851, 2.74907569056386376195936519070, 2.89111900236793762470330141692, 2.95882902574004375627074912277, 3.01810365918809937839150727925, 3.76394954278801020592040070017, 3.88682653668595991178059538036, 4.47458405778873887908822078185, 4.75911317266918689746839673193, 5.12531825747921367986760931213, 5.33331584176933187633913425429, 5.66232878047300777304000345770, 5.92073153062059669166155810561, 6.52407394181042980603428275582, 6.64009052540507893477019120956, 6.76530102180337152292914179157, 6.80822835808636887413494411107, 7.56567951674639029647512804182, 7.70316411877931333343373067411, 7.85122310648986188901733607922, 7.999341954034409427258602619972, 8.097736411986031471853808065656
