| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 16-s + 3·18-s − 6·25-s − 32-s − 3·36-s + 12·41-s − 14·49-s + 6·50-s + 64-s + 3·72-s + 4·73-s + 9·81-s − 12·82-s + 12·89-s + 14·98-s − 6·100-s + 12·113-s − 6·121-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1/4·16-s + 0.707·18-s − 6/5·25-s − 0.176·32-s − 1/2·36-s + 1.87·41-s − 2·49-s + 0.848·50-s + 1/8·64-s + 0.353·72-s + 0.468·73-s + 81-s − 1.32·82-s + 1.27·89-s + 1.41·98-s − 3/5·100-s + 1.12·113-s − 0.545·121-s + 0.0887·127-s − 0.0883·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8948107977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8948107977\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.701279446989162624981682370404, −8.059257329362955196748126272012, −7.84853874988538453576647945642, −7.48718816476406034465051724049, −6.69453020219815131350507651538, −6.41735171295850895765845842925, −5.82036882908548442716753362712, −5.53262768976432472415307376656, −4.83416586839231166768733932874, −4.23410524946504189939035172833, −3.54433173324497115703427813721, −3.01309759390232749286386603034, −2.35340232077884507145733289267, −1.71228416589805252842965220190, −0.56927382084274485641269428537,
0.56927382084274485641269428537, 1.71228416589805252842965220190, 2.35340232077884507145733289267, 3.01309759390232749286386603034, 3.54433173324497115703427813721, 4.23410524946504189939035172833, 4.83416586839231166768733932874, 5.53262768976432472415307376656, 5.82036882908548442716753362712, 6.41735171295850895765845842925, 6.69453020219815131350507651538, 7.48718816476406034465051724049, 7.84853874988538453576647945642, 8.059257329362955196748126272012, 8.701279446989162624981682370404
