| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 8·6-s + 4·7-s + 2·8-s + 6·9-s + 4·12-s + 4·13-s − 8·14-s − 4·16-s − 12·18-s + 20·19-s + 16·21-s + 8·24-s − 8·26-s − 4·27-s + 4·28-s − 4·31-s + 2·32-s + 6·36-s + 16·37-s − 40·38-s + 16·39-s − 18·41-s − 32·42-s + 10·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s − 3.26·6-s + 1.51·7-s + 0.707·8-s + 2·9-s + 1.15·12-s + 1.10·13-s − 2.13·14-s − 16-s − 2.82·18-s + 4.58·19-s + 3.49·21-s + 1.63·24-s − 1.56·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s + 0.353·32-s + 36-s + 2.63·37-s − 6.48·38-s + 2.56·39-s − 2.81·41-s − 4.93·42-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.824972467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.824972467\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 8 T^{2} + 8 T^{3} + 199 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 190 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 85 T^{2} - 18 T^{3} + 9036 T^{4} - 18 p T^{5} - 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 848 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 18 p T^{5} - 133 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 95 T^{2} - 190 T^{3} + 4 T^{4} - 190 p T^{5} - 95 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.211436171680862819264980752986, −7.76033775116971610525789240604, −7.69820320809553272417945105231, −7.50919475845530423245898231414, −7.34984464884695018596445258736, −7.21946268323579942959578671100, −6.81326868612938860292858465141, −6.40459614261678717758318989290, −6.00134730933954020510231946879, −5.67364164034654748784704880143, −5.40086759351893631576848613960, −5.32505731276368918576676120858, −5.08514616468498687969914573157, −4.57617253747426899220778771902, −4.22518941972148503764035812523, −4.00014893588602093027853337293, −3.66986786430764568360603877297, −3.30060305807911241307149981494, −3.17160584295484426980374790533, −2.65906363954321131741003417277, −2.62062605954769171436461843715, −1.84409638398667191043085923391, −1.68905030835612399919266939780, −1.14452579240521384032137373422, −0.940478734673988898420244432043,
0.940478734673988898420244432043, 1.14452579240521384032137373422, 1.68905030835612399919266939780, 1.84409638398667191043085923391, 2.62062605954769171436461843715, 2.65906363954321131741003417277, 3.17160584295484426980374790533, 3.30060305807911241307149981494, 3.66986786430764568360603877297, 4.00014893588602093027853337293, 4.22518941972148503764035812523, 4.57617253747426899220778771902, 5.08514616468498687969914573157, 5.32505731276368918576676120858, 5.40086759351893631576848613960, 5.67364164034654748784704880143, 6.00134730933954020510231946879, 6.40459614261678717758318989290, 6.81326868612938860292858465141, 7.21946268323579942959578671100, 7.34984464884695018596445258736, 7.50919475845530423245898231414, 7.69820320809553272417945105231, 7.76033775116971610525789240604, 8.211436171680862819264980752986
