L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 3·11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s + 3·17-s + 18-s + 2·19-s − 20-s − 21-s + 3·22-s + 24-s + 25-s + 2·26-s + 27-s − 28-s − 3·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.218·21-s + 0.639·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.024367186\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.024367186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.690741907054411143935588044282, −9.118286291416110457053218874732, −8.037159168746170888546014596888, −7.36406380185724711569905691147, −6.43365627177723721925713619379, −5.61014386648717677434711459948, −4.35026137440830321019845339237, −3.67080125944245417556024553165, −2.82004267261673502348315249291, −1.33162535237766037704242028947,
1.33162535237766037704242028947, 2.82004267261673502348315249291, 3.67080125944245417556024553165, 4.35026137440830321019845339237, 5.61014386648717677434711459948, 6.43365627177723721925713619379, 7.36406380185724711569905691147, 8.037159168746170888546014596888, 9.118286291416110457053218874732, 9.690741907054411143935588044282