| L(s) = 1 | − 48·41-s + 16·43-s + 16·49-s − 48·67-s − 16·83-s + 16·89-s + 32·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 7.49·41-s + 2.43·43-s + 16/7·49-s − 5.86·67-s − 1.75·83-s + 1.69·89-s + 3.09·107-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 − 4.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4430514355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4430514355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 - 8 T^{2} + 30 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 510 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + 638 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - 48 T^{2} + 1502 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 144 T^{2} + 10046 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 212 T^{2} + 28710 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.65224351498962602155126293182, −3.50500550652651940758485545068, −3.44780384459930987911864661280, −3.38136826159963122150818835432, −3.18190538955658051816260493557, −3.07177564956901429117130750718, −2.89322621922759571752112452000, −2.80052500884624663582763932713, −2.75128024370606693765584068167, −2.57015505135701579062616020344, −2.38092701988575207112190162063, −2.27583767851722544697247112880, −2.21591754775059521836641798102, −2.03378251126436996687616232777, −1.79745197764304918431076453948, −1.79543257808445929405787685318, −1.49498863075798515073389531410, −1.46531464952403834992414006378, −1.32618747164074143782458124042, −1.31530181761542214120275817810, −0.986125826142088212858678477825, −0.74957084375389549208508397428, −0.57464678855113234723805942938, −0.21767512147421907868713069862, −0.094016661724250810673702062109,
0.094016661724250810673702062109, 0.21767512147421907868713069862, 0.57464678855113234723805942938, 0.74957084375389549208508397428, 0.986125826142088212858678477825, 1.31530181761542214120275817810, 1.32618747164074143782458124042, 1.46531464952403834992414006378, 1.49498863075798515073389531410, 1.79543257808445929405787685318, 1.79745197764304918431076453948, 2.03378251126436996687616232777, 2.21591754775059521836641798102, 2.27583767851722544697247112880, 2.38092701988575207112190162063, 2.57015505135701579062616020344, 2.75128024370606693765584068167, 2.80052500884624663582763932713, 2.89322621922759571752112452000, 3.07177564956901429117130750718, 3.18190538955658051816260493557, 3.38136826159963122150818835432, 3.44780384459930987911864661280, 3.50500550652651940758485545068, 3.65224351498962602155126293182
Plot not available for L-functions of degree greater than 10.
