| L(s) = 1 | + 5-s + 5·13-s + 17-s − 9·25-s − 5·29-s − 4·37-s + 6·41-s − 3·49-s + 53-s − 9·61-s + 5·65-s + 85-s − 24·89-s − 21·101-s + 3·109-s − 6·121-s − 14·125-s + 127-s + 131-s + 137-s + 139-s − 5·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.38·13-s + 0.242·17-s − 9/5·25-s − 0.928·29-s − 0.657·37-s + 0.937·41-s − 3/7·49-s + 0.137·53-s − 1.15·61-s + 0.620·65-s + 0.108·85-s − 2.54·89-s − 2.08·101-s + 0.287·109-s − 0.545·121-s − 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.415·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.70765218763527384345011097203, −7.28375298377619756153849115959, −6.85195981893204730567768165084, −6.15376572468914799835502436507, −6.07634535574294522498473642471, −5.55146813688754187800796652057, −5.23851825975191274879847091350, −4.45189795174086928788265241562, −3.99966961187585389402326663193, −3.66009479374945347603424890642, −3.05498643090501228694564006304, −2.39759121247643680081123590756, −1.70911742946618601646113916715, −1.26178706335331558035761787869, 0,
1.26178706335331558035761787869, 1.70911742946618601646113916715, 2.39759121247643680081123590756, 3.05498643090501228694564006304, 3.66009479374945347603424890642, 3.99966961187585389402326663193, 4.45189795174086928788265241562, 5.23851825975191274879847091350, 5.55146813688754187800796652057, 6.07634535574294522498473642471, 6.15376572468914799835502436507, 6.85195981893204730567768165084, 7.28375298377619756153849115959, 7.70765218763527384345011097203
