L(s) = 1 | − 4·5-s − 2·9-s + 2·13-s + 6·25-s − 2·29-s − 5·37-s − 2·41-s + 8·45-s + 4·49-s + 14·53-s − 2·61-s − 8·65-s + 8·73-s − 5·81-s + 16·89-s + 6·97-s + 6·101-s − 8·109-s − 2·113-s − 4·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 2/3·9-s + 0.554·13-s + 6/5·25-s − 0.371·29-s − 0.821·37-s − 0.312·41-s + 1.19·45-s + 4/7·49-s + 1.92·53-s − 0.256·61-s − 0.992·65-s + 0.936·73-s − 5/9·81-s + 1.69·89-s + 0.609·97-s + 0.597·101-s − 0.766·109-s − 0.188·113-s − 0.369·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.953935566717030733901991575500, −7.65694216779119187181199716327, −7.27585552886302214491127407523, −6.75144409225339197990099963073, −6.32059922724817283548589764147, −5.69730363612505811610472063621, −5.28333946520972550880688033014, −4.75081578367799662546097597178, −4.07807451717637017208172802748, −3.79565018233219380100043453277, −3.38117068008461867828720146136, −2.73162985827577375517490103516, −1.98607535979107465010264954773, −0.912357969252171054784897597682, 0,
0.912357969252171054784897597682, 1.98607535979107465010264954773, 2.73162985827577375517490103516, 3.38117068008461867828720146136, 3.79565018233219380100043453277, 4.07807451717637017208172802748, 4.75081578367799662546097597178, 5.28333946520972550880688033014, 5.69730363612505811610472063621, 6.32059922724817283548589764147, 6.75144409225339197990099963073, 7.27585552886302214491127407523, 7.65694216779119187181199716327, 7.953935566717030733901991575500