L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 2·11-s + 12-s − 4·13-s + 2·15-s + 16-s + 17-s − 18-s + 2·19-s + 2·20-s − 2·22-s + 6·23-s − 24-s − 25-s + 4·26-s + 27-s + 8·29-s − 2·30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.288·12-s − 1.10·13-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.447·20-s − 0.426·22-s + 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s + 1.48·29-s − 0.365·30-s + 0.718·31-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
\(L(1)\) |
\(\approx\) |
\(2.242038345\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.242038345\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.421399384083873626150096127475, −7.54824214534350084432016268888, −6.91096697644343072381135554304, −6.33228258394958114991160165549, −5.32387584458447053097592654782, −4.66633364467310670079069565803, −3.42864459066443310168087753746, −2.68015210408581812080693381911, −1.88250259356155260292998911390, −0.916351569368825627287530563497,
0.916351569368825627287530563497, 1.88250259356155260292998911390, 2.68015210408581812080693381911, 3.42864459066443310168087753746, 4.66633364467310670079069565803, 5.32387584458447053097592654782, 6.33228258394958114991160165549, 6.91096697644343072381135554304, 7.54824214534350084432016268888, 8.421399384083873626150096127475