| L(s) = 1 | − 2·2-s − 3-s + 4·4-s − 5·5-s + 2·6-s − 8·8-s − 26·9-s + 10·10-s − 2·11-s − 4·12-s − 8·13-s + 5·15-s + 16·16-s − 52·17-s + 52·18-s + 26·19-s − 20·20-s + 4·22-s + 67·23-s + 8·24-s + 25·25-s + 16·26-s + 53·27-s + 69·29-s − 10·30-s − 332·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s − 0.353·8-s − 0.962·9-s + 0.316·10-s − 0.0548·11-s − 0.0962·12-s − 0.170·13-s + 0.0860·15-s + 1/4·16-s − 0.741·17-s + 0.680·18-s + 0.313·19-s − 0.223·20-s + 0.0387·22-s + 0.607·23-s + 0.0680·24-s + 1/5·25-s + 0.120·26-s + 0.377·27-s + 0.441·29-s − 0.0608·30-s − 1.92·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8022464735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8022464735\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 52 T + p^{3} T^{2} \) |
| 19 | \( 1 - 26 T + p^{3} T^{2} \) |
| 23 | \( 1 - 67 T + p^{3} T^{2} \) |
| 29 | \( 1 - 69 T + p^{3} T^{2} \) |
| 31 | \( 1 + 332 T + p^{3} T^{2} \) |
| 37 | \( 1 - 196 T + p^{3} T^{2} \) |
| 41 | \( 1 - 353 T + p^{3} T^{2} \) |
| 43 | \( 1 + 369 T + p^{3} T^{2} \) |
| 47 | \( 1 - 88 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 350 T + p^{3} T^{2} \) |
| 61 | \( 1 + 467 T + p^{3} T^{2} \) |
| 67 | \( 1 - 291 T + p^{3} T^{2} \) |
| 71 | \( 1 - 770 T + p^{3} T^{2} \) |
| 73 | \( 1 - 628 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 83 | \( 1 - 525 T + p^{3} T^{2} \) |
| 89 | \( 1 - p T + p^{3} T^{2} \) |
| 97 | \( 1 + 290 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.75851222888220370831044016785, −9.481689682394618498705545649185, −8.815507600041090726116751587567, −7.917918767373146835004163066268, −7.03849984585984076153637534974, −6.01390022844895197625069533291, −4.94319952905383978287203942787, −3.50208271960999730966444199829, −2.30264428165465967368572576511, −0.60434717543539284680372936163,
0.60434717543539284680372936163, 2.30264428165465967368572576511, 3.50208271960999730966444199829, 4.94319952905383978287203942787, 6.01390022844895197625069533291, 7.03849984585984076153637534974, 7.917918767373146835004163066268, 8.815507600041090726116751587567, 9.481689682394618498705545649185, 10.75851222888220370831044016785
