| L(s) = 1 | − 4·7-s − 9-s + 8·17-s − 8·23-s − 25-s − 8·31-s + 12·41-s − 16·47-s − 2·49-s + 4·63-s − 16·71-s − 12·73-s − 8·79-s + 81-s − 12·89-s − 28·97-s + 28·103-s + 32·113-s − 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s + 1.94·17-s − 1.66·23-s − 1/5·25-s − 1.43·31-s + 1.87·41-s − 2.33·47-s − 2/7·49-s + 0.503·63-s − 1.89·71-s − 1.40·73-s − 0.900·79-s + 1/9·81-s − 1.27·89-s − 2.84·97-s + 2.75·103-s + 3.01·113-s − 2.93·119-s + 1.63·121-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 − 0.646·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6223546656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6223546656\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.641214052552447118854817027544, −8.937392507723227211213638940724, −8.767485436494887063419299969576, −8.091336129868759217900724226901, −7.899202937869052609940500356239, −7.37914790000760467276501922256, −7.17606597733154616563819357780, −6.55596640815945222061463406359, −6.03297402500341719169067061997, −5.89309596563536002351928267567, −5.66396393363963465040847778028, −4.94519115195961000641508632514, −4.48796454073255759835434882804, −3.71639244827035964013692630335, −3.69469712571173470165046296228, −2.91086673768889109528669180180, −2.88724812615568754402538124865, −1.86768577791447890582072157859, −1.38644709173511764815421094461, −0.28752645143261348999817777872,
0.28752645143261348999817777872, 1.38644709173511764815421094461, 1.86768577791447890582072157859, 2.88724812615568754402538124865, 2.91086673768889109528669180180, 3.69469712571173470165046296228, 3.71639244827035964013692630335, 4.48796454073255759835434882804, 4.94519115195961000641508632514, 5.66396393363963465040847778028, 5.89309596563536002351928267567, 6.03297402500341719169067061997, 6.55596640815945222061463406359, 7.17606597733154616563819357780, 7.37914790000760467276501922256, 7.899202937869052609940500356239, 8.091336129868759217900724226901, 8.767485436494887063419299969576, 8.937392507723227211213638940724, 9.641214052552447118854817027544
