L(s) = 1 | + 16-s + 20·25-s + 16·67-s − 32·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
L(s) = 1 | + 1/4·16-s + 4·25-s + 1.95·67-s − 3.60·79-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.409517167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.409517167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T^{4} - 15 T^{8} - p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( 1 + 206 T^{4} + 27795 T^{8} + 206 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 734 T^{4} + 258915 T^{8} + 734 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 1234 T^{4} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 + 5582 T^{4} + 23268243 T^{8} + 5582 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2914 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.86215947201267407264216808373, −4.81802203137563566231016444833, −4.69979283645166430609544674704, −4.56223652706810401582641653739, −4.25364166050230831734022363979, −4.25015500570082797641519024672, −3.96280895594238625105097875586, −3.91718228606149615804322460251, −3.79309202577238186228373892707, −3.63316581285470163299655635133, −3.36071116789897614056847804500, −3.31555909477116528978223536273, −2.96150892377427860592915806943, −2.90773189180989679630659992956, −2.65377120116169940447090279516, −2.59253670525059862788338845986, −2.53785582265413435123503991594, −2.48022352496112040776232340583, −1.83451747351371218843714023134, −1.69346674897828024174978426636, −1.61791347771318754566543672047, −1.19292053224765028176085116692, −1.12837556752328189142159800884, −0.843942813266897962547258926178, −0.30102433333170685431838345016,
0.30102433333170685431838345016, 0.843942813266897962547258926178, 1.12837556752328189142159800884, 1.19292053224765028176085116692, 1.61791347771318754566543672047, 1.69346674897828024174978426636, 1.83451747351371218843714023134, 2.48022352496112040776232340583, 2.53785582265413435123503991594, 2.59253670525059862788338845986, 2.65377120116169940447090279516, 2.90773189180989679630659992956, 2.96150892377427860592915806943, 3.31555909477116528978223536273, 3.36071116789897614056847804500, 3.63316581285470163299655635133, 3.79309202577238186228373892707, 3.91718228606149615804322460251, 3.96280895594238625105097875586, 4.25015500570082797641519024672, 4.25364166050230831734022363979, 4.56223652706810401582641653739, 4.69979283645166430609544674704, 4.81802203137563566231016444833, 4.86215947201267407264216808373
Plot not available for L-functions of degree greater than 10.