| L(s) = 1 | + 8·5-s − 2·9-s − 4·17-s + 38·25-s − 4·29-s + 12·37-s − 4·41-s − 16·45-s + 49-s + 20·53-s − 8·61-s − 28·73-s − 5·81-s − 32·85-s + 20·89-s − 4·97-s − 24·101-s − 20·109-s + 12·113-s − 22·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 2/3·9-s − 0.970·17-s + 38/5·25-s − 0.742·29-s + 1.97·37-s − 0.624·41-s − 2.38·45-s + 1/7·49-s + 2.74·53-s − 1.02·61-s − 3.27·73-s − 5/9·81-s − 3.47·85-s + 2.11·89-s − 0.406·97-s − 2.38·101-s − 1.91·109-s + 1.12·113-s − 2·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.197869643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.197869643\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.271787274682453081680950548186, −8.676248321093563858087492291911, −8.589748372800265236248597894315, −7.51048678322729423104589438688, −7.00881059716814921000381829301, −6.28154185525062265042607550144, −6.25148198523199051609095248236, −5.61936194790385614498397188009, −5.43110360120582172209460233834, −4.80018849900442151332658226553, −4.05975063928059999821924523382, −2.73137510590455017802711867793, −2.66547510967810892791353271494, −1.94068717855646026245068222948, −1.29582036582827735540991947956,
1.29582036582827735540991947956, 1.94068717855646026245068222948, 2.66547510967810892791353271494, 2.73137510590455017802711867793, 4.05975063928059999821924523382, 4.80018849900442151332658226553, 5.43110360120582172209460233834, 5.61936194790385614498397188009, 6.25148198523199051609095248236, 6.28154185525062265042607550144, 7.00881059716814921000381829301, 7.51048678322729423104589438688, 8.589748372800265236248597894315, 8.676248321093563858087492291911, 9.271787274682453081680950548186
