| L(s) = 1 | + 3-s + 8·13-s − 4·17-s − 4·19-s + 4·23-s + 5·25-s − 27-s + 4·29-s + 8·31-s + 6·37-s + 8·39-s + 24·41-s − 8·43-s − 8·47-s − 4·51-s − 6·53-s − 4·57-s + 12·59-s − 4·61-s − 4·67-s + 4·69-s + 24·71-s − 8·73-s + 5·75-s − 16·79-s − 81-s + 8·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.21·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 25-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.986·37-s + 1.28·39-s + 3.74·41-s − 1.21·43-s − 1.16·47-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.512·61-s − 0.488·67-s + 0.481·69-s + 2.84·71-s − 0.936·73-s + 0.577·75-s − 1.80·79-s − 1/9·81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.115681547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.115681547\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.007185249115097111108239654797, −8.647414456956895549131623634688, −8.511698711921782716912394959008, −8.264866015736882378246548907461, −7.66567161028868062537591796497, −7.36502253768838549777739257030, −6.72609398785033602049816263354, −6.39996414405837680276046204512, −6.08768281995334148542535537053, −6.00009195851450917541224248759, −5.01931482159563157529158068911, −4.80047134326542298234796424976, −4.26651351612792413368214502230, −3.98012658882641848945314401863, −3.36817634719777948670928236867, −2.97562772599965526826548669934, −2.48651146526484512858525168945, −1.99663048957667265718299553985, −1.10923660635782683757001710235, −0.802156786276615521585753358598,
0.802156786276615521585753358598, 1.10923660635782683757001710235, 1.99663048957667265718299553985, 2.48651146526484512858525168945, 2.97562772599965526826548669934, 3.36817634719777948670928236867, 3.98012658882641848945314401863, 4.26651351612792413368214502230, 4.80047134326542298234796424976, 5.01931482159563157529158068911, 6.00009195851450917541224248759, 6.08768281995334148542535537053, 6.39996414405837680276046204512, 6.72609398785033602049816263354, 7.36502253768838549777739257030, 7.66567161028868062537591796497, 8.264866015736882378246548907461, 8.511698711921782716912394959008, 8.647414456956895549131623634688, 9.007185249115097111108239654797
