| L(s) = 1 | − 16·7-s − 64·25-s − 176·31-s − 36·49-s + 64·73-s + 304·79-s + 704·97-s − 112·103-s + 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 173-s + 1.02e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 2.55·25-s − 5.67·31-s − 0.734·49-s + 0.876·73-s + 3.84·79-s + 7.25·97-s − 1.08·103-s + 0.760·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s + 5.85·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.737612137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737612137\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 466 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1664 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2782 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4160 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5810 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4942 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8914 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{4} \) |
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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
−6.30849906416908693412437830873, −6.10478465648521390521766871097, −5.61362254458356761899676427107, −5.59876148754022965635710888658, −5.59257524818157604085686642532, −5.22190377123343109696560154071, −4.96031454285025270952516475346, −4.80120543012464786538411172103, −4.62112821039282335495805284946, −3.98136935092309155945015115159, −3.87198502247601285257099005230, −3.80460493862554123890398967631, −3.63980142940457781455487591228, −3.48137153723380943899903318778, −3.17523253079443008466307657615, −3.05763122170834896942337825751, −2.71470673711931664214546839366, −2.11098884714147171744501206171, −2.04985678904004030934329213547, −2.01152365581565939186641765976, −1.65404570545062085024827180489, −1.36432036659894944600216157795, −0.44524620961814616117590422170, −0.44296085759583310367773924084, −0.37610984496814733943518254230,
0.37610984496814733943518254230, 0.44296085759583310367773924084, 0.44524620961814616117590422170, 1.36432036659894944600216157795, 1.65404570545062085024827180489, 2.01152365581565939186641765976, 2.04985678904004030934329213547, 2.11098884714147171744501206171, 2.71470673711931664214546839366, 3.05763122170834896942337825751, 3.17523253079443008466307657615, 3.48137153723380943899903318778, 3.63980142940457781455487591228, 3.80460493862554123890398967631, 3.87198502247601285257099005230, 3.98136935092309155945015115159, 4.62112821039282335495805284946, 4.80120543012464786538411172103, 4.96031454285025270952516475346, 5.22190377123343109696560154071, 5.59257524818157604085686642532, 5.59876148754022965635710888658, 5.61362254458356761899676427107, 6.10478465648521390521766871097, 6.30849906416908693412437830873
