| L(s) = 1 | + 4·5-s − 9-s − 8·11-s − 4·19-s + 11·25-s − 16·29-s − 16·31-s + 4·41-s − 4·45-s + 14·49-s − 32·55-s − 16·59-s − 4·61-s + 8·79-s + 81-s + 36·89-s − 16·95-s + 8·99-s − 28·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 2.41·11-s − 0.917·19-s + 11/5·25-s − 2.97·29-s − 2.87·31-s + 0.624·41-s − 0.596·45-s + 2·49-s − 4.31·55-s − 2.08·59-s − 0.512·61-s + 0.900·79-s + 1/9·81-s + 3.81·89-s − 1.64·95-s + 0.804·99-s − 2.68·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150890724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150890724\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.498426309066702463599999221163, −9.026489002356024184429375589690, −8.954815229301843090700882632648, −8.084537019312408245267859670248, −7.85962669103444162935207870952, −7.34521936081550446993270867312, −7.27504118500016891300848084387, −6.51055679108995187652944943165, −6.03147878112757713616114365811, −5.64919098886394390211348230396, −5.57266325411591586432489183210, −5.08357540143802882808594765375, −4.79827108617241633835113521423, −3.88139454146333423112605202533, −3.59186126927450692082630023322, −2.78524995778226369577170203766, −2.47281799322469324569990449043, −1.92825156762214740300129650928, −1.72328331557520849747326333450, −0.35150413079906838597716308941,
0.35150413079906838597716308941, 1.72328331557520849747326333450, 1.92825156762214740300129650928, 2.47281799322469324569990449043, 2.78524995778226369577170203766, 3.59186126927450692082630023322, 3.88139454146333423112605202533, 4.79827108617241633835113521423, 5.08357540143802882808594765375, 5.57266325411591586432489183210, 5.64919098886394390211348230396, 6.03147878112757713616114365811, 6.51055679108995187652944943165, 7.27504118500016891300848084387, 7.34521936081550446993270867312, 7.85962669103444162935207870952, 8.084537019312408245267859670248, 8.954815229301843090700882632648, 9.026489002356024184429375589690, 9.498426309066702463599999221163
