| L(s) = 1 | − 3·5-s + 6·7-s + 6·11-s + 6·17-s + 5·25-s − 18·35-s − 24·43-s + 11·49-s − 12·53-s − 18·55-s + 12·59-s + 22·61-s − 36·67-s + 36·77-s − 18·85-s + 22·109-s − 12·113-s + 36·119-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 2.26·7-s + 1.80·11-s + 1.45·17-s + 25-s − 3.04·35-s − 3.65·43-s + 11/7·49-s − 1.64·53-s − 2.42·55-s + 1.56·59-s + 2.81·61-s − 4.39·67-s + 4.10·77-s − 1.95·85-s + 2.10·109-s − 1.12·113-s + 3.30·119-s − 0.454·121-s − 1.60·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.358895870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.358895870\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 804 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 1404 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 894 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 4110 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 155 T^{2} + 15996 T^{4} + 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 253 T^{2} + 27816 T^{4} - 253 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 12051 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 217 T^{2} + 24576 T^{4} - 217 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.33660322551234355351786961849, −6.25384774509474832836242725545, −6.19068819405059673960759778215, −5.61673595894994280939766238299, −5.56034292085955261996339090407, −5.53454965989862479457796541126, −4.95010914475644588452577792079, −4.89336054713826320562887304130, −4.81260101018612622303825195375, −4.53331293587707406749530955766, −4.37701047840381230615854579196, −4.13094447126929714585926569703, −3.81195323282954072651855807899, −3.62118755559918085025717421857, −3.38086182354188254117451582120, −3.32813640911016778759035261535, −2.90931706848926145515413634493, −2.78220779954210796095913006398, −2.14847542899726284688530348918, −1.87534219317307335857125956011, −1.64183104121081079248956001210, −1.55856239782758208204073208414, −1.18466151783060763859128630815, −0.811966546306140975447677245221, −0.37613737841077933550109035533,
0.37613737841077933550109035533, 0.811966546306140975447677245221, 1.18466151783060763859128630815, 1.55856239782758208204073208414, 1.64183104121081079248956001210, 1.87534219317307335857125956011, 2.14847542899726284688530348918, 2.78220779954210796095913006398, 2.90931706848926145515413634493, 3.32813640911016778759035261535, 3.38086182354188254117451582120, 3.62118755559918085025717421857, 3.81195323282954072651855807899, 4.13094447126929714585926569703, 4.37701047840381230615854579196, 4.53331293587707406749530955766, 4.81260101018612622303825195375, 4.89336054713826320562887304130, 4.95010914475644588452577792079, 5.53454965989862479457796541126, 5.56034292085955261996339090407, 5.61673595894994280939766238299, 6.19068819405059673960759778215, 6.25384774509474832836242725545, 6.33660322551234355351786961849
