L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 3·11-s + 12-s + 2·13-s + 16-s + 3·17-s + 2·18-s + 7·19-s − 3·22-s − 24-s − 2·26-s − 5·27-s − 6·29-s + 4·31-s − 32-s + 3·33-s − 3·34-s − 2·36-s − 8·37-s − 7·38-s + 2·39-s + 9·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 1.60·19-s − 0.639·22-s − 0.204·24-s − 0.392·26-s − 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.522·33-s − 0.514·34-s − 1/3·36-s − 1.31·37-s − 1.13·38-s + 0.320·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.680881202\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680881202\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−8.990641948152130382841694712646, −8.241263185170336936107387248232, −7.62583420866028467275275698765, −6.80760429487312719242044771491, −5.91589072886351065562126787974, −5.17711518617506807839806183328, −3.69265834103033057215511968857, −3.22985803075868355514464176224, −2.02089552612607080439032267478, −0.926909876778585217976162918755,
0.926909876778585217976162918755, 2.02089552612607080439032267478, 3.22985803075868355514464176224, 3.69265834103033057215511968857, 5.17711518617506807839806183328, 5.91589072886351065562126787974, 6.80760429487312719242044771491, 7.62583420866028467275275698765, 8.241263185170336936107387248232, 8.990641948152130382841694712646