| L(s) = 1 | − 56·17-s − 4·25-s − 56·41-s − 92·49-s + 200·73-s + 248·89-s − 584·97-s − 520·113-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.29·17-s − 0.159·25-s − 1.36·41-s − 1.87·49-s + 2.73·73-s + 2.78·89-s − 6.02·97-s − 4.60·113-s + 3.20·121-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 − 1.72·169-s + 0.00578·173-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 + 0.00473·211-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4031463865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4031463865\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3650 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 766 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4610 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9506 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{4} \) |
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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
−6.18506570500659644175479777943, −6.05745179962403504622492769541, −6.02172162805812740290092714039, −5.32018564470801238069529665326, −5.30402836287317077304614744900, −5.23722289348336981368932175379, −4.95595665870526219580467142972, −4.81446907129083433588258067280, −4.42907541494785007215318002647, −4.27214197518920437943810805142, −4.18874188670959231892465634100, −3.77288581726478830100531620783, −3.60770594327316661978364120785, −3.57355383196256195067379579185, −3.09619302499364497639361423965, −2.71775742474150711118746286608, −2.56109923961591158026576071220, −2.43850728468916313790295254397, −2.21141932682903369397591332452, −1.71213317403570871548063137943, −1.57256986404669006532199694205, −1.41442862606968520019578739808, −0.872148527331863147199875687632, −0.37922111217549063644478913260, −0.11631814328452418651769597117,
0.11631814328452418651769597117, 0.37922111217549063644478913260, 0.872148527331863147199875687632, 1.41442862606968520019578739808, 1.57256986404669006532199694205, 1.71213317403570871548063137943, 2.21141932682903369397591332452, 2.43850728468916313790295254397, 2.56109923961591158026576071220, 2.71775742474150711118746286608, 3.09619302499364497639361423965, 3.57355383196256195067379579185, 3.60770594327316661978364120785, 3.77288581726478830100531620783, 4.18874188670959231892465634100, 4.27214197518920437943810805142, 4.42907541494785007215318002647, 4.81446907129083433588258067280, 4.95595665870526219580467142972, 5.23722289348336981368932175379, 5.30402836287317077304614744900, 5.32018564470801238069529665326, 6.02172162805812740290092714039, 6.05745179962403504622492769541, 6.18506570500659644175479777943
