| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s + 3·25-s + 2·37-s + 4·41-s − 2·53-s − 4·65-s − 2·73-s − 4·85-s − 2·97-s − 2·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s + 3·25-s + 2·37-s + 4·41-s − 2·53-s − 4·65-s − 2·73-s − 4·85-s − 2·97-s − 2·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9589052527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9589052527\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.789834312403162101504104368777, −9.514494561939992989969555017758, −9.027056338595591550620069265655, −8.684468994026878119935643016065, −8.075038895783426469924256153369, −7.970727653561362937937201211123, −7.61931739906611022014648553378, −7.39065439420921194272443620685, −6.64238716791755925430759070677, −6.28416467723877955480286354843, −5.72266021236676304890805641597, −5.59502603952297633276154014009, −4.58575111085215552545235867765, −4.43913993598052819559930810697, −3.77350033148560231280538830435, −3.72863568514383898177399059257, −2.86918491009519607751869513752, −2.79545552444716794051680904709, −1.22250185924146832506956428193, −1.05868578085971676079954382992,
1.05868578085971676079954382992, 1.22250185924146832506956428193, 2.79545552444716794051680904709, 2.86918491009519607751869513752, 3.72863568514383898177399059257, 3.77350033148560231280538830435, 4.43913993598052819559930810697, 4.58575111085215552545235867765, 5.59502603952297633276154014009, 5.72266021236676304890805641597, 6.28416467723877955480286354843, 6.64238716791755925430759070677, 7.39065439420921194272443620685, 7.61931739906611022014648553378, 7.970727653561362937937201211123, 8.075038895783426469924256153369, 8.684468994026878119935643016065, 9.027056338595591550620069265655, 9.514494561939992989969555017758, 9.789834312403162101504104368777
