L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 3·11-s − 12-s + 6·13-s − 15-s + 16-s + 17-s − 18-s − 6·19-s + 20-s − 3·22-s − 2·23-s + 24-s − 4·25-s − 6·26-s − 27-s + 3·29-s + 30-s + 7·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.639·22-s − 0.417·23-s + 0.204·24-s − 4/5·25-s − 1.17·26-s − 0.192·27-s + 0.557·29-s + 0.182·30-s + 1.25·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.449283720\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449283720\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−8.383238434627910160737426894027, −7.59061352559893608017370869751, −6.62236997935134859103635097576, −6.11603948099204092508252036377, −5.78790722023797372471297470340, −4.38177170129272665669585288335, −3.88934532572653744616517678338, −2.62812632952492245504591092447, −1.60631034926281955298261074316, −0.810081456227781099994456680206,
0.810081456227781099994456680206, 1.60631034926281955298261074316, 2.62812632952492245504591092447, 3.88934532572653744616517678338, 4.38177170129272665669585288335, 5.78790722023797372471297470340, 6.11603948099204092508252036377, 6.62236997935134859103635097576, 7.59061352559893608017370869751, 8.383238434627910160737426894027