| L(s) = 1 | + 4-s + 9-s − 6·11-s − 8·19-s − 36·29-s + 2·31-s + 36-s − 6·44-s − 11·49-s + 6·59-s + 20·61-s − 64-s − 24·71-s − 8·76-s − 2·79-s + 12·89-s − 6·99-s + 36·101-s + 28·109-s − 36·116-s + 31·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 1.80·11-s − 1.83·19-s − 6.68·29-s + 0.359·31-s + 1/6·36-s − 0.904·44-s − 1.57·49-s + 0.781·59-s + 2.56·61-s − 1/8·64-s − 2.84·71-s − 0.917·76-s − 0.225·79-s + 1.27·89-s − 0.603·99-s + 3.58·101-s + 2.68·109-s − 3.34·116-s + 2.81·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.573293471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573293471\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 193 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.11981284068111357386184150054, −7.07028578368954141668226250585, −6.65203283044843767897352233788, −6.38079315795562488332181169750, −6.08485867848573934995128694992, −5.95925553933719211777171711589, −5.90794708561468377535199588299, −5.33658826514523693202033472041, −5.30125367167776825271217438093, −5.16026073419900340650664632823, −5.06123149301111033998771767411, −4.25596084819465612798249005972, −4.22758350827431420068496720724, −4.21269898922948256031232087655, −3.91146858391424674319673687106, −3.29918213439939969480557388969, −3.22631406184955385514933681809, −3.16931762337912344316148616245, −2.63371628565138776713312334180, −2.01929321309515344089331295261, −1.95339290357509476620194273383, −1.88143411264106142644343491001, −1.83520649036739363113886438926, −0.56198224361976666049644586917, −0.41142353217782761279112619525,
0.41142353217782761279112619525, 0.56198224361976666049644586917, 1.83520649036739363113886438926, 1.88143411264106142644343491001, 1.95339290357509476620194273383, 2.01929321309515344089331295261, 2.63371628565138776713312334180, 3.16931762337912344316148616245, 3.22631406184955385514933681809, 3.29918213439939969480557388969, 3.91146858391424674319673687106, 4.21269898922948256031232087655, 4.22758350827431420068496720724, 4.25596084819465612798249005972, 5.06123149301111033998771767411, 5.16026073419900340650664632823, 5.30125367167776825271217438093, 5.33658826514523693202033472041, 5.90794708561468377535199588299, 5.95925553933719211777171711589, 6.08485867848573934995128694992, 6.38079315795562488332181169750, 6.65203283044843767897352233788, 7.07028578368954141668226250585, 7.11981284068111357386184150054
