| L(s) = 1 | − 24·19-s + 8·31-s − 26·49-s + 4·61-s − 12·79-s − 128·109-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 5.50·19-s + 1.43·31-s − 3.71·49-s + 0.512·61-s − 1.35·79-s − 12.2·109-s + 2.18·121-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 − 2.61·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03582948839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03582948839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - 12 T^{2} + 23 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 17 T^{2} + 120 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 3 T + p T^{2} )^{8} \) |
| 23 | \( ( 1 + 36 T^{2} + 767 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 32 T^{2} + 183 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 72 T^{2} + 3503 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 54 T^{2} + 707 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 78 T^{2} + 2603 T^{4} - 78 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 84 T^{2} + 167 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 193 T^{2} + 27840 T^{4} + 193 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.96514174262278971480824297609, −3.92504335828855760590053390098, −3.89316031806351508350190783723, −3.87179569316202417784372084863, −3.36289399172938258528699448767, −3.32526134652412714129309451911, −3.12370890998375789071196451915, −3.02098029914625644917757263399, −2.99956394441956712294455208711, −2.89876140040363285871116094943, −2.47465147257322423390443214696, −2.43475608384686123739468764718, −2.37172855718891702475218011160, −2.26628576730449971556124734839, −2.26060717667405165336549383442, −2.13725170423137056774770422870, −1.66181054432530440957269599215, −1.62078525705690795080572163916, −1.42912919320101596940024571778, −1.31416451394707645194654086448, −1.16707506049410231481769790130, −1.16603699099775522595555848226, −0.37666979514491282527747604412, −0.33166362407911069372282534908, −0.03742473392873267428236626882,
0.03742473392873267428236626882, 0.33166362407911069372282534908, 0.37666979514491282527747604412, 1.16603699099775522595555848226, 1.16707506049410231481769790130, 1.31416451394707645194654086448, 1.42912919320101596940024571778, 1.62078525705690795080572163916, 1.66181054432530440957269599215, 2.13725170423137056774770422870, 2.26060717667405165336549383442, 2.26628576730449971556124734839, 2.37172855718891702475218011160, 2.43475608384686123739468764718, 2.47465147257322423390443214696, 2.89876140040363285871116094943, 2.99956394441956712294455208711, 3.02098029914625644917757263399, 3.12370890998375789071196451915, 3.32526134652412714129309451911, 3.36289399172938258528699448767, 3.87179569316202417784372084863, 3.89316031806351508350190783723, 3.92504335828855760590053390098, 3.96514174262278971480824297609
Plot not available for L-functions of degree greater than 10.
