| L(s) = 1 | + 5·3-s + 16·9-s − 20·13-s − 14·19-s + 35·27-s − 84·31-s + 80·37-s − 100·39-s − 100·43-s − 98·49-s − 70·57-s − 16·61-s + 90·67-s + 70·73-s − 24·79-s + 31·81-s − 420·93-s + 140·97-s − 140·103-s − 176·109-s + 400·111-s − 320·117-s − 33·121-s + 127-s − 500·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5/3·3-s + 16/9·9-s − 1.53·13-s − 0.736·19-s + 1.29·27-s − 2.70·31-s + 2.16·37-s − 2.56·39-s − 2.32·43-s − 2·49-s − 1.22·57-s − 0.262·61-s + 1.34·67-s + 0.958·73-s − 0.303·79-s + 0.382·81-s − 4.51·93-s + 1.44·97-s − 1.35·103-s − 1.61·109-s + 3.60·111-s − 2.73·117-s − 0.272·121-s + 0.00787·127-s − 3.87·129-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.826392013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.826392013\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 5 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 567 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 662 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 582 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3087 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3462 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 45 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8927 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
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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
−9.627817496455228236184166857632, −9.411495144197421748274782678632, −9.025643895099281333153523003220, −8.427273205962865973837703451892, −8.198644645007495612578767850126, −7.71499656737260742558414079910, −7.54255892290008869582082450406, −6.83015087074590882647605748021, −6.77871074613956438388261575661, −6.05261800286795867662885373551, −5.42716667676948963775412037115, −4.91000156628214671503030135763, −4.61210900887605113041925916715, −3.85373928436236758895579949847, −3.66594795170182221513517766547, −2.92745216796747528780724412230, −2.62858797602719479469188752118, −1.79835144335591125553856163411, −1.79073258657995913979332621732, −0.41090716952418475715679719021,
0.41090716952418475715679719021, 1.79073258657995913979332621732, 1.79835144335591125553856163411, 2.62858797602719479469188752118, 2.92745216796747528780724412230, 3.66594795170182221513517766547, 3.85373928436236758895579949847, 4.61210900887605113041925916715, 4.91000156628214671503030135763, 5.42716667676948963775412037115, 6.05261800286795867662885373551, 6.77871074613956438388261575661, 6.83015087074590882647605748021, 7.54255892290008869582082450406, 7.71499656737260742558414079910, 8.198644645007495612578767850126, 8.427273205962865973837703451892, 9.025643895099281333153523003220, 9.411495144197421748274782678632, 9.627817496455228236184166857632
