| L(s) = 1 | − 2-s − 2·5-s + 2·7-s − 8-s + 2·10-s − 4·11-s − 2·14-s − 16-s − 8·17-s + 4·19-s + 4·22-s + 6·23-s + 3·25-s − 2·29-s + 8·31-s + 6·32-s + 8·34-s − 4·35-s − 4·38-s + 2·40-s + 6·41-s − 6·43-s − 6·46-s + 4·47-s − 11·49-s − 3·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.534·14-s − 1/4·16-s − 1.94·17-s + 0.917·19-s + 0.852·22-s + 1.25·23-s + 3/5·25-s − 0.371·29-s + 1.43·31-s + 1.06·32-s + 1.37·34-s − 0.676·35-s − 0.648·38-s + 0.316·40-s + 0.937·41-s − 0.914·43-s − 0.884·46-s + 0.583·47-s − 1.57·49-s − 0.424·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213654786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213654786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 29 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 127 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 325 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.137161465891715757089818872249, −7.900935424269863812101493754382, −7.24141162613269232015468757221, −7.21590865983254336304879529180, −6.87153632240156676596892819072, −6.25539395746621548437920960988, −6.18116970350636880551275438435, −5.60492692809824490267219292098, −4.95870749983489333903551769528, −4.90774556585015164398633868986, −4.52478729076409106861685138615, −4.41165408108968616956331717659, −3.57864642965645565725350175716, −3.27025067724532322275270564467, −2.91930574685510432394674960332, −2.39054305983255823528738116313, −2.08874436600015990239056791943, −1.46478181898075193287977230745, −0.60293056667803174893746412237, −0.51798782905921985681394034612,
0.51798782905921985681394034612, 0.60293056667803174893746412237, 1.46478181898075193287977230745, 2.08874436600015990239056791943, 2.39054305983255823528738116313, 2.91930574685510432394674960332, 3.27025067724532322275270564467, 3.57864642965645565725350175716, 4.41165408108968616956331717659, 4.52478729076409106861685138615, 4.90774556585015164398633868986, 4.95870749983489333903551769528, 5.60492692809824490267219292098, 6.18116970350636880551275438435, 6.25539395746621548437920960988, 6.87153632240156676596892819072, 7.21590865983254336304879529180, 7.24141162613269232015468757221, 7.900935424269863812101493754382, 8.137161465891715757089818872249
