| L(s) = 1 | + 2·7-s + 10·13-s + 2·19-s + 2·25-s − 10·31-s − 2·37-s + 2·43-s − 11·49-s + 4·61-s − 16·67-s + 4·73-s + 2·79-s + 20·91-s + 34·97-s − 16·103-s + 34·109-s − 10·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2.77·13-s + 0.458·19-s + 2/5·25-s − 1.79·31-s − 0.328·37-s + 0.304·43-s − 1.57·49-s + 0.512·61-s − 1.95·67-s + 0.468·73-s + 0.225·79-s + 2.09·91-s + 3.45·97-s − 1.57·103-s + 3.25·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.641892416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.641892416\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−8.823148884472770583686001033090, −8.292632570800360480618355268794, −7.889909888006925658409255709715, −7.72253562023634107256833095808, −7.22073864149760325444080616349, −6.81574589779543637368539971185, −6.28554038131584619415916166866, −6.19601898208036417515526512565, −5.63429678017844926628142228106, −5.42599137125184929601519370235, −4.83706904053187060669113718097, −4.58022826876496116172597702043, −3.81573367725641187139774016798, −3.80632793632654084893354210457, −3.23592837460017240011479082399, −2.92573440025065869437037121154, −1.96026140301677575807151011176, −1.72304635608116323224123903823, −1.23579631315049212244451551519, −0.59702866500578487456240907865,
0.59702866500578487456240907865, 1.23579631315049212244451551519, 1.72304635608116323224123903823, 1.96026140301677575807151011176, 2.92573440025065869437037121154, 3.23592837460017240011479082399, 3.80632793632654084893354210457, 3.81573367725641187139774016798, 4.58022826876496116172597702043, 4.83706904053187060669113718097, 5.42599137125184929601519370235, 5.63429678017844926628142228106, 6.19601898208036417515526512565, 6.28554038131584619415916166866, 6.81574589779543637368539971185, 7.22073864149760325444080616349, 7.72253562023634107256833095808, 7.889909888006925658409255709715, 8.292632570800360480618355268794, 8.823148884472770583686001033090
